Rational Cubic Curves
Problems with cubics
- Cannot do exact conics (circles, ellipses)
- proof: how to write x(u) = r cos(u) as a cubic? You can’t!
- Not invariant under perspective transformations
- projection of a cubic is not a cubic
- Q: why is this a problem?
- A: can’t just project control points - must project entire curve
Solution: homogeneous coordinates
- Define 4 cubics: X(u), Y(u), Z(u) and W(u)
- Curve is now: x(u) = X(u)/W(u), etc…
The curve is now a rational function
Non-Uniform Rational B–Splines (NURBS) widely used in computer-aided geometric design (CAGD)