Geometric Modeling91.580.201

Curves (continued)

Cubic Bezier and B-Spline CurvesFarin Chapters 4-5, 6, 8Mortenson Chapters 4, 5

4 Typical Types of Parametric Curves

• Interpolating– Curve passes through all control points.

• Hermite– Defined by its 2 endpoints and tangent vectors at endpoints.– Interpolates all its control points.– Not invariant under affine transformations.– Special case of Bezier and B-Spline.

• Bezier*– Interpolates first and last control points.– Invariant under affine transformations.– Curve is tangent to first and last segments of control polygon.– Easy to subdivide.– Curve segment lies within convex hull of control polygon.– Variation-diminishing.– Special case of B-spline.

• B-Spline*– Not guaranteed to interpolate control points.– Invariant under affine transformations.– Curve segment lies within convex hull of control polygon.– Variation-diminishing.– Greater local control than Bezier.

Control points influence curve shape.

source: Mortenson, source: Mortenson, AngelAngel

*focus of this lecture

Notation

• Mortenson– Control points and parameter:

• Farin– Control points and parameter: R and ,, 3

10 tnbbb

R and ,, 310 unppp

source: Mortenson, Farinsource: Mortenson, Farin

Bezier CurvesFarin Chapter 4

The de Casteljau Algorithm

Parabola Construction

• Change of notation from Chapter 3 (previously):

• Now:

• Add:

• By substitution:

source: Farinsource: Farin

10)1(],0[ bbb ttt 21)1(],1[ bbb ttt

1010 )1()( bbb ttt

2111 )1()( bbb ttt

)()()1()( 11

10

20 ttttt bbb

22

1022

0 )1(2)1()( bbbb ttttt

The de Casteljau Algorithm

• Given:

• set:

• Example: cubic

rni

nr

,0

,1

source: Farinsource: Farin

)()()1()( 11

1 ttttt ri

ri

ri

bbbii t bb )(0

R and ,, 310 tnbbb

30

21

123

20

112

101

0

bbbb

bbb

bb

b

curve point

Conceptually elegant, but not necessarily the fastest algorithm…

source: Farinsource: Farin

21b

Some Bezier Curve Properties

• Affine invariance: Linear interpolation is affinely invariant.• Invariance under affine parameter transformations (beyond

[0,1] to [a,b]):

• Convex hull: Curve stays within convex hull of control polygon.– Each intermediate point is a convex barycentric combination of

previously generated points.– Planar control polygon generates planar curve.– Helpful for intersection tests.

• Endpoint interpolation: Curve passes through b0 and bn.– Verify using scheme for t =0 and t =1.

)()()( 11

1 uab

auu

ab

ubu r

iri

ri

bbb

source: Farinsource: Farin

Bezier CurvesFarin Chapter 5-6

& Mortenson Chapter 4The Bernstein Form of a Bezier Curve

Bezier

• Geometric form (cubic case):

3

03, )()(

iii uBu pp

source: Mortensonsource: Mortenson

iii uu

iuB

3

3, )1(3

)(

Evaluate at u=0 and u=1 to show tangents related to first and last control polygon line segment.

33

22

12

03 )1(3)1(3)1()( ppppp uuuuuuu

33

223

123

023 )33()363()133()( ppppp uuuuuuuuuu

32

22

12

02 3)69()3129()363()( ppppp uuuuuuuuu

Bernstein polynomials.

4 = number of control points = degree + 1

)(3)0( 01 ppp u )(3)1( 23 ppp u

Bezier

• Geometric form (general case):

n

iini uBu

0, )()( pp

source: Mortensonsource: Mortenson

inini uu

i

nuB

)1()(,

Bernstein polynomials.

n+1 = number of control points = degree + 1

Rational form is invariant under perspective transformation:where hi are projective space coordinates (weights)

n

inii

n

iinii

uBh

uBhu

0,

0,

)(

)()(

pp

See Chapter 13 of Farin for rational Bezier material.

More Bezier Curve Properties

• Symmetry: Bernstein polynomials are symmetric with respect to u and 1- u.

• Convex Hull (again):

source: Farin, Mortensonsource: Farin, Mortenson

Convex combination, so Bezier curve points all lie within convex hull of control polygon.

n

ini uB

0, 1)(

Bezier curve with 4 control points

inini uu

i

nuB

)1()(,

More Bezier Curve Properties

• Degree elevation leaving curve unchanged

inini tt

i

ntB

)1()(

source: Farinsource: Farin

Adding a control point elevates degree by 1.

1,,0for )1(

1)1( 1

)1(

nin

i

n

iiii bbb

New vertices b(1)i obtained from old polygon by piecewise linear

interpolation at parameter values i/(n+1).

New control polygon is in convex hull of old one.

More Bezier Curve Properties• Repeated degree elevation: (in the limit) produces control polygon

sequence that converges to the actual curve.

• Variation Diminishing:– Piecewise linear interpolation is variation diminishing.– Degree elevation uses piecewise linear interpolation.– Each successive control polygon cannot intersect a given plane more often

than the control polygon upon which it is based.– Thus, the curve cannot intersect the plane more often than any of the control

polygons.– Corollary: convex control polygon produces convex curve segment.

source: Farinsource: Farin

Composite Bezier Curves

Evaluate at u=0 and u=1 to show tangents related to first and last control polygon line segment.

)(3)0( 01 ppp u )(3)1( 23 ppp u

Joining adjacent curve segments is an alternative to degree elevation.

source: Mortensonsource: Mortenson

Collinearity of cubic Bezier control points produces G1 continuity at join

point:

For G2 continuity at join point, 5 vertices must be coplanar.

B-Spline CurvesMortenson Chapter 5

& Farin Chapter 8

B-Spline

n

iiKi uNu

0, )()( pp

otherwise 0)(

if 1)(

1,

11,

uN

tutuN

i

iii

• Geometric form (non-uniform, non-rational case), where K controls degree (K -1) of basis functions:

• Cubic B-splines can provide C2 continuity at curve segment join points.

Convex combination, so B-spline curve points all lie within convex hull of control polygon.

n

iKi uN

0, 1)(

Rational form (NURBS) is invariant under perspective transformation, where hi are projective space coordinates (weights).

n

iKii

n

iiKii

uNh

uNhu

0,

0,

)(

)()(

pp

source: Mortensonsource: Mortenson

ti are n+1+K knot values that relate u to the control points.

Uniform case: space knots at equal intervals of u.

Repeated knots move curve closer to control points.

N

N

N N

N

N N

N N

See Chapter 13 of Farin for rational B-spline material.

1

1,1

1

1,,

)()()()()(

iki

kiki

iki

kiiki tt

uNut

tt

uNtuuN

Some B-Spline Curve Properties• Affine invariance: Linear interpolation is affinely invariant.• Invariance under affine parameter transformations (beyond

[0,1] to [a,b]):• Convex hull: Curve stays within convex hull of control

polygon (see later slide).– Each intermediate point is a convex barycentric combination of

previously generated points.– Planar control polygon generates planar curve.– Helpful for intersection tests.

• No endpoint interpolation: unlike Bezier• Variation diminishing• Transformation to some other parametric forms:

– B-Spline curve can be transformed into Bezier form.• See Mortenson p. 170.

– B-Spline curve can be transformed into Hermite form.• See Mortenson p. 170-171.

source: Farin, Mortensonsource: Farin, Mortenson

Cubic B-Spline • Geometric form (cubic case):

320 6

1

6

4

6

1)1( pppp

source: Mortensonsource: Mortenson

Evaluate at u=0 and u=1 to show curve endpoints and tangent directions related to control polygon.

32

22

12

02

2

1)123(

2

1)43(

2

1)12(

2

1)( ppppp uuuuuuuuu

B-spline cubic basis functions

4 = number of control points = degree + 1

20 2

1

2

1)0( ppp u

n

iiKi uNu

0, )()( pp

N

N N

N N otherwise 0)(

if 1)(

1,

11,

uN

tutuN

i

iii

1

1,1

1

1,,

)()()()()(

iki

kiki

iki

kiiki tt

uNut

tt

uNtuuN

Convex combination, so B-spline curve points all lie within convex hull of control polygon.

n

iKi uN

0, 1)(

33

223

123

023

6

1)1333(

6

1)463(

6

1)133(

6

1)( ppppp uuuuuuuuuu

Uniform cubic B-spline:N1,4

N2,4 N3,4N4,4

210 6

1

6

4

6

1)0( pppp

31 2

1

2

1)1( ppp u

More B-Spline Curve Properties

• Symmetry: Basis functions possess parametric symmetry:

source: Farin, Mortensonsource: Farin, Mortenson

Example: Uniform cubic B-spline:

33

223

123

023

6

1)1333(

6

1)463(

6

1)133(

6

1)( ppppp uuuuuuuuuu

N1,4N2,4 N3,4 N4,4

)1()( ,1, uNuN KiKKi

Shape Characterization of Uniform Cubic B-Spline Curve Segment

• Shape depends on control polygon• 3 (non-collinear) control points determine

triangle• Type of curve segment and its monotonicity

depends on location of 4th point p3 relative to triangle p0, p1, p2:

– SP: spiral control polygon induces convex, monotone curve segment.

– U: U-shaped convex control polygon induces convex, monotone curve segment.

– V:V-shaped convex control polygon induces 1 inflection point at curve segment endpoint. Curve segment is convex, monotone.

– N: N-shaped control polygon induces1 inflection point with convex, monotone pieces on both sides of inflection point.

– SI: self-intersecting control polygon induces 1 or 2 monotone curve subsegments.

– D: self-intersecting control polygon may induce loop, cusp or 2 inflection points.

– In all cases, curve segment can be partitioned into at most 2 monotone pieces.

p0

p1

p2

SPSP

SP

SPSP

SI

UV

V

N

N

N

SP

D

source: Daniels, et al. building on Wang et al.source: Daniels, et al. building on Wang et al.

The deBoor Algorithm

• Given: knots & control points

& parameter u, generalize de Casteljau algorithm.

• Example: quadratic case with 4 knots and 3 control points

source: Farinsource: Farin],[],[],[

],[],[

],[

232

121

10

uuuuuu

uuuu

uu

bbb

bb

b

curve point

Conceptually elegant, but not necessarily the fastest algorithm…

source: Farinsource: Farin

32110 ],[],,[ uuuu bb

,, 10 uu

],[],[],[ 212

11

12

2 uuuu

uuuu

uu

uuuu bbb

],[],[ 2102

010

02

2

12

2 uuuu

uuuu

uu

uu

uu

uubb

],[],[ 3213

121

13

3

12

1 uuuu

uuuu

uu

uu

uu

uubb

knot sequence u0, u1, u2, u3 = 0, 1, 3, 4 and u = 2.0

Geometric Modeling91.580.201

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