1. CS175 2003 1 CS 175 – Week 8 Bézier Curves Definition, Algorithms

2. CS175 2003 2 Overview polynomial curves monomial basis Lagrange basis Bernstein basis

3. CS175 2003 3 Polynomial Curves Weierstrass Approximation Theorem model curves with polynomials monomial basis evaluation with Horner’s method coefficients do not relate to shape Lagrange basis coefficients are interpolation points

4. CS175 2003 4 Neville’s Algorithm “repeated” linear interpolation “dynamic” evaluation procedure Lagrange basis functions with Neville’s algorithm up recurrence relation down recurrence relation

5. CS175 2003 5 De Casteljau’s Algorithm simplify Neville’s algorithm repeated linear evaluation with constant weights gives approximating limit curve up and down recurrence defines basis functions

6. CS175 2003 6 Bernstein Basis properties non-negativity symmetry partition of unity linear independence boundary values derivative and maximum

7. CS175 2003 7 Bézier Curves properties convex hull affine invariance symmetry endpoint interpolation variation diminishing property uniform convergence