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

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

CS 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

CS 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

CS 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

CS 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

CS 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