The diagram shows a geometric construction for a cubic spline curve.

The curve is parametrized by t.

Given A, B, C, D. the following constraints are set:

AE = t AB ; BF = t BC; CG = t CD; EH = t EF; FI = t FG; HJ = t HI

The curve created by tracing J as t goes from 0 to 1 has the following properties:

It is tangent to AB at A.

It lies inside the convex hull of A,B,C,D. (The convex hull is the smallest convex polygon containing the points)

In fact the curve is a cubic in t. Can you use a computer algebra system to work out its coefficients in terms of the coordinates of A, B, C, D?