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Representation of Curves amp Surfaces
Prof Lizhuang Ma
Shanghai Jiao Tong University
Contents
bull Specialized Modeling Techniques
bull Polygon Meshes
bull Parametric Cubic Curves
bull Parametric Bi-Cubic Surfaces
bull Quadric Surfaces
bull Specialized Modeling Techniques
The Teapot
Representing Polygon Meshes
bull Explicit representationbull By a list of vertex coordinates
bull Pointers to a vertex listbull Pointers to an edge list
))()()(( 222111 nnn zyxzyxzyxP
Pointers to A Vertex List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)324(
)421(
2
1
P
P
Pointers to An Edge List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)(
)(
)(
)(
)(
)(
)(
4322
5411
1145
21244
2433
2322
1211
EEEP
EEEP
PVVE
PPVVE
PVVE
PVVE
PVVE
Parametric Cubic Curves
bull The cubic polynomials that define a curve segment are of the form
10)(
)(
)(
23
23
23
tdtctbtatz
dtctbtaty
dtctbtatx
zzzz
yyyy
xxxx
TtztytxtQ )]()()([)(
Parametric Cubic Curves
bull The curve segment can be rewrite as
bull Where
zzzz
yyyy
xxxx
T
dcba
dcba
dcba
C
tttT 123
TCtztytxtQ T )]()()([)(
Parametric Cubic Curves
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()(
TttCTCdt
d]123[ 2
Tzzzyyyxxx ctbtactbtactbta ]232323[ 222
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Contents
bull Specialized Modeling Techniques
bull Polygon Meshes
bull Parametric Cubic Curves
bull Parametric Bi-Cubic Surfaces
bull Quadric Surfaces
bull Specialized Modeling Techniques
The Teapot
Representing Polygon Meshes
bull Explicit representationbull By a list of vertex coordinates
bull Pointers to a vertex listbull Pointers to an edge list
))()()(( 222111 nnn zyxzyxzyxP
Pointers to A Vertex List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)324(
)421(
2
1
P
P
Pointers to An Edge List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)(
)(
)(
)(
)(
)(
)(
4322
5411
1145
21244
2433
2322
1211
EEEP
EEEP
PVVE
PPVVE
PVVE
PVVE
PVVE
Parametric Cubic Curves
bull The cubic polynomials that define a curve segment are of the form
10)(
)(
)(
23
23
23
tdtctbtatz
dtctbtaty
dtctbtatx
zzzz
yyyy
xxxx
TtztytxtQ )]()()([)(
Parametric Cubic Curves
bull The curve segment can be rewrite as
bull Where
zzzz
yyyy
xxxx
T
dcba
dcba
dcba
C
tttT 123
TCtztytxtQ T )]()()([)(
Parametric Cubic Curves
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()(
TttCTCdt
d]123[ 2
Tzzzyyyxxx ctbtactbtactbta ]232323[ 222
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
The Teapot
Representing Polygon Meshes
bull Explicit representationbull By a list of vertex coordinates
bull Pointers to a vertex listbull Pointers to an edge list
))()()(( 222111 nnn zyxzyxzyxP
Pointers to A Vertex List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)324(
)421(
2
1
P
P
Pointers to An Edge List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)(
)(
)(
)(
)(
)(
)(
4322
5411
1145
21244
2433
2322
1211
EEEP
EEEP
PVVE
PPVVE
PVVE
PVVE
PVVE
Parametric Cubic Curves
bull The cubic polynomials that define a curve segment are of the form
10)(
)(
)(
23
23
23
tdtctbtatz
dtctbtaty
dtctbtatx
zzzz
yyyy
xxxx
TtztytxtQ )]()()([)(
Parametric Cubic Curves
bull The curve segment can be rewrite as
bull Where
zzzz
yyyy
xxxx
T
dcba
dcba
dcba
C
tttT 123
TCtztytxtQ T )]()()([)(
Parametric Cubic Curves
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()(
TttCTCdt
d]123[ 2
Tzzzyyyxxx ctbtactbtactbta ]232323[ 222
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Representing Polygon Meshes
bull Explicit representationbull By a list of vertex coordinates
bull Pointers to a vertex listbull Pointers to an edge list
))()()(( 222111 nnn zyxzyxzyxP
Pointers to A Vertex List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)324(
)421(
2
1
P
P
Pointers to An Edge List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)(
)(
)(
)(
)(
)(
)(
4322
5411
1145
21244
2433
2322
1211
EEEP
EEEP
PVVE
PPVVE
PVVE
PVVE
PVVE
Parametric Cubic Curves
bull The cubic polynomials that define a curve segment are of the form
10)(
)(
)(
23
23
23
tdtctbtatz
dtctbtaty
dtctbtatx
zzzz
yyyy
xxxx
TtztytxtQ )]()()([)(
Parametric Cubic Curves
bull The curve segment can be rewrite as
bull Where
zzzz
yyyy
xxxx
T
dcba
dcba
dcba
C
tttT 123
TCtztytxtQ T )]()()([)(
Parametric Cubic Curves
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()(
TttCTCdt
d]123[ 2
Tzzzyyyxxx ctbtactbtactbta ]232323[ 222
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Pointers to A Vertex List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)324(
)421(
2
1
P
P
Pointers to An Edge List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)(
)(
)(
)(
)(
)(
)(
4322
5411
1145
21244
2433
2322
1211
EEEP
EEEP
PVVE
PPVVE
PVVE
PVVE
PVVE
Parametric Cubic Curves
bull The cubic polynomials that define a curve segment are of the form
10)(
)(
)(
23
23
23
tdtctbtatz
dtctbtaty
dtctbtatx
zzzz
yyyy
xxxx
TtztytxtQ )]()()([)(
Parametric Cubic Curves
bull The curve segment can be rewrite as
bull Where
zzzz
yyyy
xxxx
T
dcba
dcba
dcba
C
tttT 123
TCtztytxtQ T )]()()([)(
Parametric Cubic Curves
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()(
TttCTCdt
d]123[ 2
Tzzzyyyxxx ctbtactbtactbta ]232323[ 222
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Pointers to An Edge List
)( 4321 VVVVV
))()(( 444111 zyxzyx
)(
)(
)(
)(
)(
)(
)(
4322
5411
1145
21244
2433
2322
1211
EEEP
EEEP
PVVE
PPVVE
PVVE
PVVE
PVVE
Parametric Cubic Curves
bull The cubic polynomials that define a curve segment are of the form
10)(
)(
)(
23
23
23
tdtctbtatz
dtctbtaty
dtctbtatx
zzzz
yyyy
xxxx
TtztytxtQ )]()()([)(
Parametric Cubic Curves
bull The curve segment can be rewrite as
bull Where
zzzz
yyyy
xxxx
T
dcba
dcba
dcba
C
tttT 123
TCtztytxtQ T )]()()([)(
Parametric Cubic Curves
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()(
TttCTCdt
d]123[ 2
Tzzzyyyxxx ctbtactbtactbta ]232323[ 222
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Parametric Cubic Curves
bull The cubic polynomials that define a curve segment are of the form
10)(
)(
)(
23
23
23
tdtctbtatz
dtctbtaty
dtctbtatx
zzzz
yyyy
xxxx
TtztytxtQ )]()()([)(
Parametric Cubic Curves
bull The curve segment can be rewrite as
bull Where
zzzz
yyyy
xxxx
T
dcba
dcba
dcba
C
tttT 123
TCtztytxtQ T )]()()([)(
Parametric Cubic Curves
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()(
TttCTCdt
d]123[ 2
Tzzzyyyxxx ctbtactbtactbta ]232323[ 222
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Parametric Cubic Curves
bull The curve segment can be rewrite as
bull Where
zzzz
yyyy
xxxx
T
dcba
dcba
dcba
C
tttT 123
TCtztytxtQ T )]()()([)(
Parametric Cubic Curves
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()(
TttCTCdt
d]123[ 2
Tzzzyyyxxx ctbtactbtactbta ]232323[ 222
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Parametric Cubic Curves
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()(
TttCTCdt
d]123[ 2
Tzzzyyyxxx ctbtactbtactbta ]232323[ 222
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Tangent Vector
Ttzdt
dty
dt
dtx
dt
dtQtQ
dt
d)]()()([)()(
TttCTCdt
d]123[ 2
Tzzzyyyxxx ctbtactbtactbta ]232323[ 222
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
ContinuityBetween Curve Segments
bull G0 geometric continuityndash Two curve segments join together
bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)
of the two segmentsrsquo tangent vectors are equal at a join point
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
ContinuityBetween Curve Segments
bull C1 continuousndash The tangent vectors of the two cubic curve segments
are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710
bull Cn continuousndash The direction and magnitude of through
the nth derivative are equal at the joint point
)]([ tQdtd nn
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
ContinuityBetween Curve Segments
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
ContinuityBetween Curve Segments
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Three Types ofParametric Cubic Curves
bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent
vectors
bull Beacutezier Curvesndash Defined by two endpoints and two control points w
hich control the endpointrsquo tangent vectors
bull Splinesndash Defined by four control points
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Parametric Cubic Curves
bull Representationbull Rewrite the coefficient matrix as
ndash where M is a 4x4 basis matrix G is called the geometry matrix
ndash So
1)(
)(
)(
)(2
3
44342414
43332313
42322212
41312111
4321t
t
t
mmmm
mmmm
mmmm
mmmm
GGGG
tz
ty
tx
tQ
TCtQ )(
MGC
4 endpoints or tangent vectors
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Parametric Cubic Curves
ndash Where is called the blending functions
BGTMGtQ )(
TMB
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Hermite Curves
bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4
bull What isndash Hermite basis matrix MH
ndash Hermite geometry vector GH
ndash Hermite blending functions BH
bull By definition
GH=[P1 P4 R1 R4]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Hermite Curves
bull Since THH
THH
THH
THH
MGPQ
MGRQ
MGPQ
MGPQ
0123)1(
0100)0(
1111)1(
1000)0(
4
1
4
1
0011
1110
2010
3010
4141 HHH MGRRPPG
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Hermite Curves
bull So
bull And
0011
0121
0032
1032
0011
1110
2010
30101
HM
TH tttttttttB 23232323 232132
HHHH BGTMGtQ )(
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that
bull What is1048710ndash Beacutezier basis matrix MB
ndash Beacutezier geometry vector GB
ndash Beacutezier blending functions BB
Beacutezier Curves
)(3)1(
)(3)0(
34
4
12
1
PPQR
PPQR
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Beacutezier Curves
bull By definitionbull Then
bull So
HBB MGPPPP
3011
3000
0300
0301
4321
4321 PPPPGB
4141 RRPPGH
TMMGTMGtQ HHBBHH )()(
TMGTMMG BBHHBB )(
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Beacutezier Curves
bull And
HBBHHBB MGMMM
0001
0033
0363
1331
43
32
22
13 )1(3)1(3)1()( PtPttPttPttQ
TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Convex Hull
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t
he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i
nterfere with rapid interactive reshaping of a curve
Spline
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
B-Spline
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments
bull Uniformndash The knots are spaced at equal intervals of the paramet
er t1048710bull Non-rational
ndash Not rational cubic polynomial curves
Uniform NonRational B-Splines
3 10 mPPP m
mQQQ 43
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix
bull If bull Then
Uniform NonRational B-Splines
iiii PPPP 223
miPPPPG iiiiBSi 3123
Tiiii ttttttT 1)()()( 23
1)( iiiBsBsii tttTMGtQ
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
bull So B-Spline basis matrix
bull B-Spline blending functions
Uniform NonRational B-Splines
TBs tttttttB 323233 1333463)1(6
1
0001
1333
4063
1331
6
1BsM
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
bull The knot-value sequence is a nondecreasing sequence
bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)
bull So
Uniform NonRational B-Splines
)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Uniform NonRational B-Splines
bull Where is j th-order blending function for weighting control point Pi
)()()(
)()()(
)()()(
0
1)(
3114
43
34
2113
32
23
1112
21
12
11
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
tBtt
tttB
tt
tttB
otherwise
ttttB
iii
ii
ii
ii
iii
ii
ii
ii
iii
ii
ii
ii
iii
)( tB ji
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Knot Multiplicity amp Continuity
bull Since is within the convex hull of and
bull If is within the convex hull of and and the convex hull of and so it will lie on
bull If will lie on bull If will lie on both and and th
e curve becomes broken
)( itQ 23 ii PP 1iP
)(1 iii tQtt 23 ii PP
1iP 12 ii PP1iP
12 ii PP
)(21 iiii tQttt 1iP
)(321 iiiii tQtttt 1iP iP
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity
Knot Multiplicity amp Continuity
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
NURBSNonUniform Rational B-Splines
bull Rational bull and are defined as the ratio of two cubi
c polynomialsbull Rational cubic polynomial curve segments are rati
os of polynomials
bull Can be Beacutezier Hermite or B-Splines
)()( tytx
)(
)()(
tW
tXtx
)(tz
)(
)()(
tW
tYty
)(
)()(
tW
tZtz
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so
me path then
bull Since are cubicsbull where
SMtGtGtGtGtsQ )()()()(( 4321)=
TMGtQ )(
SMGtQ )(
)(tGi
TMGtGi )( 4321 iiiii ggggG=
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Parametric Bi-Cubic Surfacesbull So
SM
gggg
gggg
gggg
gggg
MTtsQ TT
44342414
43332313
42322212
41313211
)(
10 tsSMGMT TT
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Hermite Surfaces
SMGMTtsQ HHTH
T )( TMGtRtRtPtP HH )()()( 4141
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Beacutezier Surfaces
SMGMTtsQ BsBsTBs
T )(
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Normals to Surfaces
Ss
MGMTtsQs
TT
)(
TTT ssMGMT 0123 2
SMGMTt
tsQs
TT
)()(
SMGMss TT 0123 2
)()( tsQt
tsQs
normal vector
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Quadric Surfaces
bull Implicit surface equation
bull An alternative representation
bull with
0 PQPT
0222222)( 22
2 kjzhygxfxzeyzdxyczbyaxzyxf
1
z
y
x
P
kjhg
jcef
hebd
gfda
Q
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another
Quadric Surfaces
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Fractal Models
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]
Grammar-Based Models
bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]