1. CS175 2003 1 CS 175 – Week 9 B-Splines Definition, Algorithms

2. CS175 2003 2 Overview the de Boor algorithm B-spline curves B-spline basis functions B-spline algorithms uniform B-splines and subdivision

3. CS175 2003 3 The De Boor Algorithm modify the de Casteljau algorithm start with different blossom values gives approximating limit curve down recurrence gives another polynomial basis neighbouring curve segments join smoothly

4. CS175 2003 4 B-Spline Curves piecewise polynomial C n-  continuous at  -fold knots local control affine invariance local convex hull property interpolate n-fold control point interpolate control point at n-fold knot variation diminishing

5. CS175 2003 5 Knot Insertion add local detail > refine curve increase degree refine knot vector add one knot replace n-1 c.p.’s with n new c.p.’s Boehm’s algorithm one level of de Boor’s algorithm conversion to piecewise Bézier

6. CS175 2003 6 B-Spline Basis Functions recursive definition piecewise polynomial C n-  continuous at  -fold knots compact support partition of unity non-negativity basis for all piecewise polynomials recursive formula for derivative

7. CS175 2003 7 Uniform B-Splines knots are equally spaced basis functions are just shifted convolution theorem subdivision insert all “mid-knots” n=2 > Chaikin’s corner cutting general n > Lane-Riesenfeld