B Spline Blending Functions 1 Bk 1 t 0 tk t tk 1 otherwise t tk Bk d 1 t Bk d t tk d 1 tk tk d t Bk 1 d 1 t tk d tk 1 The recurrence relation starts with the 1st order B splines just boxes and builds up successively higher orders This algorithm is the Cox de Boor algorithm Carl de Boor is in the CS department here at Madison Uniform Cubic B splines Uniform cubic B splines arise when the knot vector is of the form 3 2 1 0 1 n 1 Each blending function is non zero over a parameter interval of length 4 All of the blending functions are translations of each other Each is shifted one unit across from the previous one Bk d t Bk 1 d t 1 The blending functions are the result of convolving a box with itself d times although we will not use this fact 3 t t 1 0 0 8 0 0 6 0 2 0 4 0 2 0 0 4 0 2 0 4 0 2 0 8 0 4 0 8 0 6 1 1 t 0 8 B 2 1 1 2 B 0 1 1 4 1 2 1 1 6 1 2 2 0 2 1 8 0 4 2 2 0 8 2 4 0 6 B1 1 t 1 0 8 2 6 1 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 2 2 2 4 2 6 1 2 8 0 6 B3 1 t 3 2 8 B0 1 t 1 2 3 1 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 2 2 2 4 2 6 2 8 B2 1 t Bk 1 B 1 1 1 2 0 6 0 4 0 2 0 0 t B 3 1 0 6 t 1 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 2 2 2 4 2 6 2 8 3 B2 2 t 3 t 1 2 0 8 1 0 6 0 4 0 2 0 1 t 3 B0 2 t 1 t 1 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 0 8 0 1 2 B 0 2 1 4 0 2 1 6 0 2 2 0 4 1 8 0 4 2 2 0 8 2 4 1 2 6 1 2 2 8 B1 2 t 0 6 3 1 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 2 2 2 4 2 6 2 8 B0 2 t Bk 2 B 1 2 1 2 1 0 8 0 6 t B 2 2 3 t 2 2 t 1 Bk 3 B 0 3 B 1 3 0 8 0 8 0 7 0 7 0 6 0 4 t t 3 2 1 B0 3 t 2t 2 6t 3 2 2 t 3 t 2 2 t 1 1 t 0 1 0 8 0 6 0 4 0 0 2 0 2 0 4 0 6 1 t 0 8 1 2 1 4 1 6 2 1 8 2 2 1 0 8 0 6 0 4 0 0 2 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 2 1 8 2 2 0 2 4 0 2 6 0 1 2 8 0 2 0 1 2 4 0 3 0 2 2 6 0 3 0 5 3 0 4 2 8 B1 3 t 0 5 3 B0 3 t 0 6 t 1 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 2 2 2 4 2 6 2 8 3 B0 4 t B0 4 B 0 4 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 B0 4 t 3 3 1 3t 3 15t 2 21t 5 B0 4 t 6 3t 3 3t 2 3t 1 1 t 3 3 t 2 2 t 1 1 t 0 0 t 1 Uniform Cubic B spline Blending Funcs 0 7 B0 4 B1 4 B2 4 B3 4 B4 4 B5 4 B6 4 0 6 0 5 0 4 0 3 0 2 0 1 3 2 7 2 3 2 1 6 1 3 0 9 0 6 0 2 0 1 0 5 0 8 1 2 1 5 1 9 2 2 2 6 2 9 3 3 3 6 4 4 3 4 7 0 t Computing the Curve n X t Pk Bk 4 t k 0 0 25 0 2 0 15 P0B0 4 0 1 P1B1 4 P4B4 4 P2B2 4 P6B6 4 P3B3 4 0 05 P5B5 4 t 4 7 4 4 3 3 6 2 9 3 3 2 6 1 9 2 2 1 5 1 2 0 5 0 8 0 1 0 6 0 2 0 9 1 6 1 3 2 2 3 3 2 7 0 Using Uniform B splines At any point t along a piecewise uniform cubic Bspline there are four non zero blending functions Each of these blending functions is a translation of B0 4 Consider the interval 0 t 1 We pick up the 4th section of B0 4 We pick up the 3rd section of B1 4 We pick up the 2nd section of B2 4 We pick up the 1st section of B3 4 Blending Function on 0 1 0 7 0 6 B1 4 x t P0 B0 4 t P1B1 4 t P2 B2 4 t P3 B3 4 t P0 1 3t 3t 2 t 3 1 P1 4 6t 2 3t 3 2 3 P 1 3 t 3 t 3 t 6 2 P3 t 3 B2 4 0 5 0 4 0 3 0 2 B0 4 0 1 B3 4 t 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 Uniform B spline on 0 1 Four control points are required to define the curve for 0 t 1 The blending functions sum to one and are positive everywhere The curve lies inside its convex hull Does the curve interpolate its endpoints Look at the blending functions to decide There is also a matrix form for the curve 1 x t P0 6 P1 P2 1 3 3 3 6 0 P3 3 3 3 0 0 1 1 t 3 4 t 2 1 t 0 1 Uniform B spline at …
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