In the mathematical field of numerical analysis and in computer graphics, a Bézier spline is a spline curve where each polynomial of the spline is in Bézier form.
In other words, a Bézier spline is simply a series of Bézier curves stacked end to end where the last point of one curve coincides with the starting point of the next curve. Usually cubic Bézier curves are used, and additional control points (called handles) are added to define the shape of each curve.
Definition
Given a spline S of degree n with k knots xi 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-2}(x) := & \sum_{\nu=0}^{n} \beta_{\nu,k-2} b_{\nu,n}(x - x_{k -2}) & x \in [x_{k-2}, x_{k-1}] \\
\end{matrix}\right.](http://wpcontent.answers.com/math/a/e/d/aed618e06f8f78567291370c3e739853.png)
Examples
 Eight-segment quadratic Bézier spline (red) approximating a circle (black) with control points |
 Four-segment cubic Bézier spline (red) approximating a circle (black) with control points |
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