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Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016

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divided differences of the product of two functions, h(t) = f(t)g(t) h[t 0, t 1,…,t k ] = f[t 0, t 1, …,t r ] g[t r, t r+1, …,t k ] = f[t 0 ]g[t 0, t 1, …,t k ] + f[t 0, t 1 ]g[t 1, …,t k ] + … + f[t 0, t 1, …,t k 1 ]g[t k 1,t k ] + f[t 0, t 1, …,t k ] g[t k ] h k (t j ; t) = (t j t) k 1 + = (t j t) k 2 + (t j t) + = h k 1 (t j ; t) (t j t) +

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h k [t i k,..., t i ;t ] = h k 1 [t i k,…, t i 1 ;t] + h k 1 [t i k,…, t i ;t](t i t) k th divided difference of (t j t) k 1 + : B-spline M k, i (t) M k-1,i-1 (t)

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h k [t i k,..., t i ;t ] = h k 1 [t i k,…, t i 1 ;t] + h k 1 [t i k,…, t i ;t](t i t) M k, i (t)M k 1,i 1 (t) + {h k 1 [t i k+1,…, t i ;t ] h k 1 [t i k,…, t i 1 ;t]} = {M k 1, i (t) M k 1, i 1 (t)} M k, i (t) + = M k 1,i 1 (t) M k, i (t) = M k 1,i 1 (t) + M k 1, i (t) similar to the de Casteljau’s algorithm repeated linear interpolation is performed between two consecutive splines a table to construct splines may also be generated

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t i k, t i k+1 M 1,i k+1 (t) M 2,i k+2 (t) t i k+1, t i k+2 M 1,i k+2 (t) M 2,i k+3 (t) t i k+2, t i k+3 M 1,i k+3 (t) Mk1,i1(t)Mk1,i1(t) Mk,i(t)Mk,i(t) Mk1,i(t)Mk1,i(t) t i 3, t i 2 M1,i2(t)M1,i2(t) M2,i1(t)M2,i1(t) t i 2, t i 1 M1,i1(t)M1,i1(t) M2,i(t)M2,i(t) t i 1, t i M1,i(t)M1,i(t)

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M k, i (t) is non-zero in the knot span t i k t t i and zero elsewhere M 1, i (t) is non-zero only in one span, t i 1 t t i M 1, i (t) is constant in t i 1 t t i can be computed using the standardization condition for t [t i 1, t i ) = 0 elsewhere

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for t [t i 1, t i ) = 0 elsewhere M k, i (t) = M k 1,i 1 (t) + M k 1, i (t) for t [t i k, t i ) = 0 elsewhere

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0 10 1 0.5 2 56 M 1,1 M 1,2 M 1,3 M 1,4 01 23 4 M 2,2 M 2,3 M 2,4 M 3,3 M 3,4 M 4,4

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