CPSC 424 Rational Curves Syllabus - Ugrad.cs.ubc.ca

CPSC 424 

Rational Curves 

© Alla Sheffer& Wolfgang Heidrich 

Syllabus 

Curves in 2D and 3D 

• Implicit vs. Explicit vs. Parametric curves 

• Bézier curves 

• Continuity 

• B-Splines 

• Rational curves 

• Subdivision curves 

Properties of Curves and Surfaces 

• Differential Geometry 

Surfaces 

© Alla Sheffer & Wolfgang Heidrich 

1


Motivation 

• Conic sections: 

– All curves generated by the intersection of a cone 

with a plane 

– Parabolae, hyperbolae, ellipses/circles 

• Bézier and B-Spline curves cannot represent all conic 

sections 

– Parabolae possible, but not hyperbolae and 

ellipses/circles 

– We have seen a hint of this in assignment 2 for the 

example of circles… 



Motivation (cont) 

• Bézier and B-Spline curves are invariant under affine 

transformations but not projective transformations 

• Example: canonical perspective transformation 

Fx 

( t) / Fz 

( t) 

Fx 

( t) 

 

 

 

Fy 

( t) / Fz 

( t) 

 

Fy 

( t) 

 

1/ F ( t) 

1 

z 

 

1 Fz 

( t) 

 

1 

Fx 

( t) 

 

 

 

Fy 

( t) 

 

1 

F 

( t) 

 

z 

 

1 

division on right side cannot be represented with the 

de Casteljau or de Boor algorithm 

 

 

 

 

 

1 

1 


2


Definition: 

• A rational curve is of the form 

F( 

t) : 

m 

 

i0 

m 

B 

 

i0 

m 

i 

B 

( t) 

w 

m 

i 

i 

( t) 

w 

b 

i 

i 

 

B 

m m 

i 

m 

i0 

 

m 

B 

j 

j0 

( t) 

w 

i 

( t) 

w 

j 

b 

i 

w 

0, 

w0 w 1 

with weights 

Note: 

i 

m 

• We can use either of Bézier and B-Spline basis here 



In homogeneous coordinates: 

• Note that we get the original Bezier 

curves if all weights are 1! 

• Clear from this representation: 

the projective image of a rational 

curve is again a rational curve 

(projective invariance) 

 

 

 

 

 

F( 

t) : 

 

 

 

 

 

m 

 

i0 

m 

 

i0 

m 

 

i0 

m 

B 

B 

B 

 

i0 

m 

i 

m 

i 

m 

i 

B 

( t) 

w 

( t) 

w 

( t) 

w 

m 

i 

i 

i 

i 

z 

( t) 

w 

x 

y 

i 

 

 

 

 

 

 

 

 

 

 

 

i 

i 

i 


3


Other properties: 

• Convex hull 

• Curve interpolates start and endpoint (rational Bézier 

only) 

• De Casteljau, de Boor and other algorithms now have 

to be performed in homogeneous coordinates! 


OpenGL 

void gluNurbsCurve( GLUnurbsObj *nobj, GLint nknots, 

GLfloat *knot, GLint stride, GLfloat *ctlarray, GLint order, 

GLenum type ) 

nobj - pointer to object created by gluNewNurbsRenderer 

nknots - number of knots 

knot – knot vector 

stride - “width” = dimension 

ctlarray - array of control points. 

order - degree+1 

type - enum of types 


4

OpenGL - example 

GLfloat knots[8] = { 

0.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 1.0 

}; 

gluBeginCurve(nurb); 

gluNurbsCurve(nurb, 8, knots, 3, &ctlpoints[0][0], 

4, GL_MAP1_VERTEX_3); 

gluEndCurve(nurb); 


Subdivision Curves 

Represent smooth curve by approximating 

polyline 

At the limit = curve 

Each iteration 

• Add new points (~double) 

• Approximating - Move old points 

• Interpolating – Keep old points 


5

Bezier Subdivision 

To create complex curve need to explicitly 

enforce continuity – not very useful 


Corner Cutting 


6





7





8





9



Corner Cutting – Chaikin Algorithm 

control point 

limit curve 

control polygon 

Limit – quadratic B-spline curve 


10

Cubic B-Spline (corner cutting) 

3/4 

1/8 1/8 


Cubic Corner Cutting 


11

The 4-point scheme 




12





13





14





15





16





17





18




control point 

limit curve 

control polygon 


19

Subdivision curves 

Non interpolatory subdivision schemes 

• Corner Cutting 

Interpolatory subdivision schemes 

• The 4-point scheme 


20

CPSC 424 Rational Curves Syllabus - Ugrad.cs.ubc.ca

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