Bezier curve: equation and properties

A Bezier curve is a parametric curve defined by a set of control points forming a control polygon. The curve approximates the polygon: it starts at the first and ends at the last control point, and is pulled towards the other points. It was developed by Pierre Bezier at Renault.

Equation

For $n+1$ control points $\mathbf B_0,\dots,\mathbf B_n$ (degree $n$):

$$\mathbf P(u)=\sum_{i=0}^{n}\mathbf B_i\,J_{n,i}(u),\qquad 0\le u\le1$$

with the Bernstein basis

$$J_{n,i}(u)=\binom{n}{i}u^i(1-u)^{n-i}=\frac{n!}{i!(n-i)!}u^i(1-u)^{n-i}$$

For a cubic ($n=3$): $J_{3,0}=(1-u)^3$, $J_{3,1}=3u(1-u)^2$, $J_{3,2}=3u^2(1-u)$, $J_{3,3}=u^3$.

Properties

1. The curve is of degree one less than the number of control points.
2. It passes through the first and last control points: $\mathbf P(0)=\mathbf B_0$, $\mathbf P(1)=\mathbf B_n$.
3. End tangents lie along the first and last polygon sides: $\mathbf P'(0)=n(\mathbf B_1-\mathbf B_0)$, $\mathbf P'(1)=n(\mathbf B_n-\mathbf B_{n-1})$.
4. Convex hull property: the curve lies inside the convex hull of its control points, because the basis functions are non-negative and sum to 1.
5. Variation diminishing: the curve does not cross any straight line more times than the control polygon does.
6. Symmetry: reversing the order of the points gives the same curve.
7. Invariant under affine transformation: transform the control points and redraw.
8. Global control: moving one control point changes the whole curve.

Limitations

• Degree rises with the number of points, so many points give a high-degree, costly curve.
• No local control.
• Cannot represent exact conics such as a circle (rational Bezier is required).
• Joining segments smoothly needs the three points around the joint to be collinear (tangent, $G^1$, continuity); for $C^1$ continuity between segments of equal degree they must also be equally spaced.