list_cubic_spline

Cubic Interpolation

Cubic interpolation (Bezier, Hermite, Cardinal, B-Spline) written in list comprehension and vectorization, no Numpy! Used to draw trajectories between points.

Curve trajectories are all given by

$$A \times B \times C$$ $$A = \begin{bmatrix} 1 & t & t^2 & t^3 \end{bmatrix}$$

Bezier

$$B \times C = \begin{bmatrix} 1 & 0 & 0 & 0 \\\ -3 & 3 & 0 & 0 \\\ 3 & -6 & 3 & 0 \\\ -1 & 3 & -3 & 1 \end{bmatrix} \begin{bmatrix} P_1 \\\ H_1 \\\ H_2 \\\ P_2 \end{bmatrix}$$

Hermite

$$B \times C = \begin{bmatrix} 1 & 0 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ -3 & 3 & -2 & -1 \\\ 2 & -2 & 1 & 1 \end{bmatrix} \begin{bmatrix} P_1 \\\ P_2 \\\ H_1 \\\ H_2 \end{bmatrix}$$

Cardinal

$$B \times C = \begin{bmatrix} 0 & 1 & 0 & 0 \\\ -s & 0 & s & 0 \\\ 2s & s-3 & 3-2s & -s \\\ -s & 2-s & s-2 & s \end{bmatrix} \begin{bmatrix} P_1* \\\ P_2 \\\ P_3 \\\ P_4* \end{bmatrix}$$

B-Spline

$$B \times C = \begin{bmatrix} 1 & 4 & 1 & 0 \\\ -3 & 0 & 3 & 0 \\\ 3 & -6 & 3 & 0 \\\ -1 & 3 & -3 & 1 \end{bmatrix} \begin{bmatrix} H_1 \\\ H_2 \\\ H_3 \\\ H_4 \end{bmatrix}$$

Variable Description

$s$ is the tension, and can vary from 0 to 1. 0 is linear interpolation, 0.5 is Catmull-Rom
$P$ denotes control points that lie on the curve
$H$ denotes control points that don't lie on the curve (handles)
(*) starred denotes points that aren't interpolated