Basis 2 producing curves by b splines the basis

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Unformatted text preview: e into a set of knot spans [ti, ti+1) The B-splines can be defined by * Bspline Basis Bspline Basis (2) Producing Curves by B-Splines • The basis functions are multiplied to the control points and to define arbitrary curves 0 10/10/2008 t 3 Lecture 5 m+1 Knots • The knots produce a vector that defines the domain of the curve • The knots must be in the increasing order • But not necessarily uniform • If uniformly sampled and the degree is 3 uniform cubic bspline 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Knots Here is an example of non-uniform knots http://i33www.ira.uka.de/applets/mocca/html/noplugin/BSplineBasis/AppBSplineBasis/index.html Some Terms • Order k: the number of control points that affect the sampled value • Degree k-1 (the basis functions are polynomials of degree k-1) • Control points Pi (i=0,…,m) • Knots : tj, (j=0,…, n) • An important rule : n – m = k • The domain of function tk-1 ≦ t ≦tm+1 – Below, k = 4, m = 9, domain, t3 ≦ t ≦t10 0 t 3 m+1 Clamped Bsplines • The first and last knot values are repeated with multiplicity equal to the order (= degree + 1) • The end points pass the control point • For cubic Bsplines, the multiplicity of the first / last knots must be 4 (repeated four times) Controlling the shape of B-splines • Moving the control points is...
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This document was uploaded on 03/26/2014.

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