CPSC 424 Rational Curves Syllabus - Ugrad.cs.ubc.ca
CPSC 424 Rational Curves Syllabus - Ugrad.cs.ubc.ca
CPSC 424 Rational Curves Syllabus - Ugrad.cs.ubc.ca
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<strong>Rational</strong> <strong>Curves</strong><br />
Motivation<br />
• Conic sections:<br />
– All curves generated by the intersection of a cone<br />
with a plane<br />
– Parabolae, hyperbolae, ellipses/circles<br />
• Bézier and B-Spline curves <strong>ca</strong>nnot represent all conic<br />
sections<br />
– Parabolae possible, but not hyperbolae and<br />
ellipses/circles<br />
– We have seen a hint of this in assignment 2 for the<br />
example of circles…<br />
© Alla Sheffer & Wolfgang Heidrich<br />
<strong>Rational</strong> <strong>Curves</strong><br />
Motivation (cont)<br />
• Bézier and B-Spline curves are invariant under affine<br />
transformations but not projective transformations<br />
• Example: <strong>ca</strong>noni<strong>ca</strong>l perspective transformation<br />
Fx<br />
( t) / Fz<br />
( t)<br />
Fx<br />
( t)<br />
<br />
<br />
<br />
Fy<br />
( t) / Fz<br />
( t)<br />
<br />
Fy<br />
( t)<br />
<br />
1/ F ( t)<br />
1 <br />
z<br />
<br />
1 Fz<br />
( t)<br />
<br />
1<br />
Fx<br />
( t)<br />
<br />
<br />
<br />
Fy<br />
( t)<br />
<br />
1<br />
F<br />
( t)<br />
<br />
z<br />
<br />
1 <br />
division on right side <strong>ca</strong>nnot be represented with the<br />
de Casteljau or de Boor algorithm<br />
<br />
<br />
<br />
<br />
<br />
1<br />
1<br />
© Alla Sheffer & Wolfgang Heidrich<br />
2
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