720 likes | 725 Vues
Least Squares Curves, Rational Representations, Splines and Continuity. Dr. Scott Schaefer. Degree Reduction. Given a set of coefficients for a Bezier curve of degree n +1, find the best set of coefficients of a Bezier curve of degree n that approximate that curve. Degree Reduction.
E N D
Least Squares Curves, Rational Representations, Splines and Continuity Dr. Scott Schaefer
Degree Reduction • Given a set of coefficients for a Bezier curve of degree n+1, find the best set of coefficients of a Bezier curve of degree n that approximate that curve
Degree Reduction • Problem: end-points are not interpolated
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
The PseudoInverse • What happens when isn’t invertible?
Constrained Least Squares Optimization Solution Constraint Space Error Function F(x)
Degree Reduction • Problem: end-points are not interpolated
Rational Curves • Curves defined in a higher dimensional space that are “projected” down
Rational Curves • Curves defined in a higher dimensional space that are “projected” down
Rational Curves • Curves defined in a higher dimensional space that are “projected” down
Rational Curves • Curves defined in a higher dimensional space that are “projected” down
Why Rational Curves? • Conics
Why Rational Curves? • Conics
Why Rational Curves? • Conics
Why Rational Curves? • Conics