Anupam Saxena Associate Professor Indian Institute of Technology KANPUR

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) +

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)

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

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) Mk1,i1(t)Mk1,i1(t) Mk,i(t)Mk,i(t) Mk1,i(t)Mk1,i(t) t i  3, t i  2 M1,i2(t)M1,i2(t) M2,i1(t)M2,i1(t) t i  2, t i  1 M1,i1(t)M1,i1(t) M2,i(t)M2,i(t) t i  1, t i M1,i(t)M1,i(t)

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

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

M 1,1 M 1,2 M 1,3 M 1, M 2,2 M 2,3 M 2,4 M 3,3 M 3,4 M 4,4