Given N points
Find Nice-looking curve through these points
Polynomial Interpolation 1 polynomial of degree N-1 matches points
piece together N-1 cubic curves together smoothly
Gaussian Elimination can do
Let the curves be S as described as fallows,
we have the properties:
to show how (2) stands：
so, , such that and note that:
With the two properties above, all we have to do is to determine Pi through the first differential equations:
which leads us to simple matrix computation.
For Example, let n = 7, we can construct a matrix bellow:
Solve , then plug them to Si to finish the approach.
Time complexity: T=O(N)