CS 175 – Week 9 B-Splines Definition, Algorithms

# CS 175 – Week 9 B-Splines Definition, Algorithms

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## CS 175 – Week 9 B-Splines Definition, Algorithms

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1. CS 175 – Week 9B-SplinesDefinition, Algorithms

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

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. B-Spline Curves • piecewise polynomial • Cn- 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. 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. B-Spline Basis Functions • recursive definition • piecewise polynomial • Cn- continuous at -fold knots • compact support • partition of unity • non-negativity • basis for all piecewise polynomials • recursive formula for derivative

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