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

\begin{array}{r} x_{t} $ = $ \sum_{i = 0}^{n} (\binom{n}{i}) (1 - t)^{n - i} t^{i} a_{i} \\ y_{t} $ = $ \sum_{i = 0}^{n} (\binom{n}{i}) (1 - t)^{n - i} t^{i} b_{i} . \end{array}

where

(\binom{n}{i}) = \frac{n!}{(n - i)! i!}