The formula for a Bezier curve


Suppose we have $n+1$ control points $P_0 =(a_0, b_0),$ $P_1=(a_1, b_1),$ ... $P_ n=(a_ n, b_ n).$ We will represent the corresponding Bézier curve by points $(x_ t,y_ t)$ where $t$ runs from 0 to 1. In other words, for each $t$ between 0 and 1 we get a point $(x_ t, y_ t)$ and together these points form the curve. The formulae for $x_ t$ and $y_ t$ are

  $\displaystyle x_ t $=$ \sum _{i=0}^ n { n \choose i} (1-t)^{n-i} t^ i a_ i  $    
  $\displaystyle y_ t $=$ \sum _{i=0}^ n { n \choose i} (1-t)^{n-i} t^ i b_ i. $    

where

  \[ { n \choose i} = \frac{n!}{ (n-i)! i!} \]    
is the binomial coefficient.

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