De Casteljau's algorithm

In the mathematical field of numerical analysis, De Casteljau's algorithm is a recursive method to evaluate polynomials in Bernstein form or Bézier curves, named after its inventor Paul de Casteljau. De Casteljau's algorithm can also be used to split a single Bézier curve into two Bézier curves at an arbitrary parameter value.

Although the algorithm is slower for most architectures when compared with the direct approach, it is more numerically stable.

A Bézier curve B (of degree n, with control points ${\displaystyle \beta _{0},\ldots ,\beta _{n}}$) can be written in Bernstein form as follows

where b is a Bernstein basis polynomial

The curve at point t₀ can be evaluated with the recurrence relation

Then, the evaluation of ${\displaystyle B}$ at point ${\displaystyle t_{0}}$ can be evaluated in ${\displaystyle n}$ steps of the algorithm. The result ${\displaystyle B(t_{0})}$ is given by :

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