Bézier spline

In the mathematical subfield of numerical analysis and in computer graphics a Bézier spline is a spline curve where each polynomial of the spline is in Bézier form. A Bézier spline is sometimes called bezigon as Bézier splines are like polygons but instead of straight lines they consist of Bézier curves.

Bezigons can be made big or small or rotated with no "jaggies". Adobe PostScript fonts are made up of bezigons. Bezigons can be used to create scalable shapes directly.

Definition

Given a spline S of degree n with k knots x_{i_{we can write the spline as a Bézier spline as}}

$S(x) := \left\{ \begin{matrix} S_0(x) := & \sum_{\nu=0}^{n} \beta_{\nu,0} b_{\nu,n}(x) & x \in [x_0, x_1) \\ S_1(x) := & \sum_{\nu=0}^{n} \beta_{\nu,1} b_{\nu,n}(x - x_1) & x \in [x_1, x_2) \\ \vdots & \vdots \\ S_{k-1}(x) := & \sum_{\nu=0}^{n} \beta_{\nu,k-1} b_{\nu,n}(x - x_{k -1}) & x \in [x_{k-1}, x_k) \\ \end{matrix}\right.$

Categories: Numerical analysis | Curves