This page is intended to be a brief reference on implementing Bézier curves. It is not intended to be a full discussion of the topic, only a reference.
A Bézier curve is a sequence of control points on a parameter interval.
The control points may be scalars or vectors, and there may be an number of them; we will denote the control points as p_0, p_1, \dots, p_n. The n here is the order
of the Bézier curve and is one less than the number of control points.
We will refer to the parameter interval as going from t_0 to t_n. We assume t_0 < t_n.
To find the point on a Bézier curve at some parameter value t, we proceed as follows.
Let t_a by the relative parameter value defined by t_a = \frac{t-t_0}{t_n-t_0}; let t_b = 1-t_a.
If n=0, return p_0. Otherwise, define a set of control points where the new n is the old n-1 and the new p_i is the old t_b p_i + t_a p_{i+1}; repeat until the new n is zero.
In addition to providing the point on the curve at t, De Casteljau’s algorithm also splits the original Bézier curve into two: the set of all p_0 (in the order created) define the portion of the Bézier curve in the interval [t_0, t] and the set of all p_n (in the reverse order created) define the portion of the Bézier curve in the interval [t, t_n]
At t = t_0, the value of a Bézier curve is p_0. At t = t_n, the value of a Bézier curve is p_n. In general, the Bézier curve does not pass through any of its other control points.
Bézier curves always remain inside the convex hull of their control points.
Within the interval t_0 \le t \le t_n, de Casteljau’s algorithm is unconditionally numerically stable: it gives the value of the polynomial with as much numerical precision as the control points and t values are themselves specified. Outside that interval de Casteljau’s algorithm still works in principle, but it is not stable, accumulating numerical error at a rate polynomial in the distance outside the interval.
Bézier curves also can be degree-elevated and degree-reduced; can efficiently determine their derivative using a hodograph; can represent conic sections if the control points are homogeneous coordinates; and can be generated in a smooth spline using B-splines and their variants. For an extensive treatment, see Thomas Sederberg’s book Computer Aided Geometric Design.