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

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

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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

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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

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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

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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

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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

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