CS CS 175 – Week 8 Bézier Curves Definition, Algorithms

CS Overview polynomial curves monomial basis Lagrange basis Bernstein basis

CS 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

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

CS 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

CS Bernstein Basis properties non-negativity symmetry partition of unity linear independence boundary values derivative and maximum

CS Bézier Curves properties convex hull affine invariance symmetry endpoint interpolation variation diminishing property uniform convergence