Catmull-Rom spline - Definition

In mathematics, a Catmull-Rom spline is a cardinal spline with a tension of 0.5.

Given n+1 points

p₀, ..., p_n,

to be interpolated with n cubic Hermite curve segments, for each curve we have a starting point p_i and an ending point p_i+1 with starting tangent m_i and ending tangent m_i+1 with the tangents defined by

<math> \mathbf{m}_i = \frac{1}{2}(\mathbf{p}_{i+1}-\mathbf{p}_{i-1}) <math>.

with the first and last tangent given.