Bezier curves expressed parametrically as products of matrices.

Question

Express the point P(t) on the Bezier curve on the control points $P0 = (5, 3)$, $P1 = (1, 8)$, $P2 = (7, 4)$ a product of three matrices.

Formula: $$A(kj) = (-1)^{k-j} \frac{k!}{j!(k-j)!} \frac{ L!}{k!(L-k)!}$$ when $k \ge j$ and $0$ otherwise.

Answer 1

I have no idea what the symbols in your question mean, so this is just a guess.

The point $P(t)$ can be obtained by the de Casteljau algorithm. Each step of this algorithm is just computing a convex combination of previous points, and can therefore be expressed as a matrix multiplication. Specifically:

1. Take the three original control points (as a column vector).

2. Multiply by a $2 \times 3$ matrix to get two points for the first stage of the de Casteljau algorithm.

3. Multiply by a $1 \times 2$ matrix to get the final point $P(t)$ on the curve.