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Graphics II 91.547

B-SplinesNURBS

Session 3A

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B-splines

Suppose you wanted C0, C1 and C2 continuity at curve boundaries.

pi

pi3

pi1 pi2

Ci

Use all four control points to determine boundary continuitiesand only require that the curve pass “close” to the points.

C p bi i kk

ku u( ) ( )

0

3

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B-splines: Sharing of Control Points

pi

pi3

pi1 pi2

Ci

Ci1pi4

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B-splines: Using continuity requirements tocompute geometry matrix/blending functions

pi

pi3

pi1 pi2

Ci

Ci1pi4

C p b C p bi i kk

k i i kk

k( ) ( ) ( ) ( )1 1 0 00

3

1 10

3

C0 continuity here requires:

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B-splines: Using continuity requirements tocompute geometry matrix/blending functions

pi

pi3

pi1 pi2

Ci

Ci1pi4

bb bb bb b

b

0

1 0

2 1

3 2

3

1 01 01 01 0

0 0

( )( ) ( )( ) ( )( ) ( )

( )

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B-splines: Using continuity requirements tocompute geometry matrix/blending functions

bb bb bb b

b

0

1 0

2 1

3 2

3

1 01 01 01 0

0 0

'( )'( ) '( )'( ) '( )'( ) '( )

'( )

bb bb bb b

b

0

1 0

2 1

3 2

3

1 01 01 01 0

0 0

"( )"( ) "( )"( ) "( )"( ) "( )

"( )

Similarly, the C1 and C2 continuity conditions give:

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B-spline blending functions

b0b3

b1 b2

b u u ub u ub u u ub u

016

2 3

116

2 3

216

2 3

316

3

1 3 34 6 31 3 3 3

( )( )( )

23

0 1

M BS

16

1 4 1 03 0 3 0

3 6 3 01 3 3 1

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B-splines:Local versus global parameter

Ci Ci1 Ci2Ci 1

pi pi1 pi2 pi3 pi4

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B-splines:Recursively defined basis functions

]...[

)()(

otherwise0 if1

321

1

1,1

1

1,,

11,

k

iki

kiki

iki

kiiki

iii

tttt

ttuBut

ttuBtu

uB

tutuB

For any “knot vector”:

Order i

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First order basis functions:

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Second order basis functions:

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Knot Vectors

Only Requirement:

Image: David Rogers

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Definition of B Spline Curve

1

1

)()(n

iiik uBu pp

k Order of the spline

1n Number of control points

1 nk Number of knots in knot vector *

* Notation according to D.F. Rogers

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Knot Vectors:Open, Uniform

Result: spline passes through end control vertices

Image: David Rogers

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Building Up Basis Functions

Image: David Rogers

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Methods of Control

0 Change number and/or position of control vertices0 Change order k0 Change type of knot vector

- Open uniform- Open non uniform

0 Use multiple coincident control vertices0 Use multiple internal knot values

Image: David Rogers

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Control: Change Order

Image: David Rogers

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Control: Non Uniform Knot Vectors

Image: David Rogers

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Control: Knot Vector Type

Image: David Rogers

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Control:Multiple Coincident Vertices

Image: David Rogers

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Control: Duplicate Knot Values

Image: David Rogers

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Rational B-Splines (NURBS)

i

ii

ii

ii

i

i

i

hzhyhxh

zyx

1

Equivalency betweenHomogeneous representations:

1

1

1

1

)(

)()( n

iiik

n

iiiik

huB

xhuBuxDoing the perspective

division gives:

Interpreted as “weighting factor” for control verticesih

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NURBSEffect of weighting factor

Image: David Rogers

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Drawing NURBS in OpenGL

GLUnurbsObj *curveName;

curveName = gluNewNurbsRenderer();gluBeginCurve (curveName);gluNurbsCurve (curveName, nknots, *knotVector,

stride, *ctrlPts, degParam, GL_MAP1_VERTEX_3);gluEndCurve (curveName);

See OpenGL Programming Guide Ch. 12 for details ofusing the glu NURBS interface

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NURBS:Code Example

120 goto 120

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Extending from Curves to Surfaces

)()(),(1

1

1

1,

vBuBvuvv

u

u

v

vuuvu dk

n

k

n

kdkkk

pP

•Cartesian product of B-Spline basis functions

•Order can be different for u and v directions