Properties of Cubic Spline

Spline:

In Computer Graphics, the term spline curves refer to any composite curve formed with polynomial sections satisfying specified boundary conditions at the section endpoints. The concept of the mathematical spline curve used in computer-aided geometric design is derived from the physical industrial spline.
Depending on the type of polynomial used and the set of boundary conditions satisfied, there are mainly three types of spline curves used in Graphics Applications.

1. Piecewise Cubic Spline
2. Bezier Spline
3. B-Spline

Cubic B-Spline:

Considering P(t) as the parametric position vector of any point along the curve, the general expression for a B-Spline curve of degree d-1 is given by,

Where P_i are the input set of n+1 control points and B_i,d(t) are the n+1 B-Spline blending functions defined by the Cox-deBoor recursion formula as,