bundle block adjustment with 3d natural cubic splines

From 0th continuity,y 0 = a 00 + a 01 x 0 + a 02 x 2 0 + a 03 x 3 0y 1 = a 00 + a 01 x 1 + a 02 x 2 1 + a 03 x 3 1y 1 = a 10 + a 11 x 1 + a 12 x 2 1 + a 13 x 3 1y 2 = a 10 + a 11 x 2 + a 12 x 2 2 + a 13 x 3 2· · ·y n−1 = a (n−1)0 + a (n−1)1 x n−1 + a (n−1)2 x 2 n−1 + a (n−1)3 x 3 n−1y n = a (n−1)0 + a (n−1)1 x n + a (n−1)2 x 2 n + a (n−1)3 x 3 nFrom 1st continuity,a 01 + 2a 02 x 1 + 3a 03 x 2 1 = a 11 + 2a 12 x 1 + 3a 13 x 2 1a 11 + 2a 12 x 2 + 3a 13 x 2 2 = a 21 + 2a 22 x 2 + 3a 23 x 2 2· · ·a (n−2)1 + 2a (n−2)1 x n−1 + 3a (n−2)3 x 2 n−1 = a (n−1)1 + 2a (n−1)2 x n−1 + 3a (n−1)3 x 2 n−1From 2nd continuity,2a 02 + 6a 03 x 1 = 2a 12 + 6a 13 x 12a 12 + 6a 13 x 2 = 2a 22 + 6a 23 x 2· · ·2a (n−2)2 + 6a (n−2)3 x n−1 = 2a (n−1)2 + 6a (n−1)3 x n−1From boundary conditions,2a 01 + 6a 03 x 0 = 02a (n−1)2 + 6a (n−1)3 x n = 038