# Bézier curve

Given a set of $n+1$ control points $P_0$, $P_1$, ..., $P_n$, the corresponding Bézier curve (or Bernstein-Bézier curve) is given by $$ \textbf{C}(t)=\sum_{i=0}^n \textbf{P}_i B_{i,n}(t), $$ where $B_{i,n}(t)$ is a Bernstein polynomial and t in [0,1].

#### Rational Bezier curve

A "rational" Bézier curve is defined by $$ C(t)=(sum_(i=0)^(n)B_(i,p)(t)w_iP_i)/(sum_(i=0)^(n)B_(i,p)(t)w_i), $$

## Bernstein polynormials

Wolfram. $$ B_{i,n}(t)=\binom{n}{i} t^i (1-t)^{n-i} $$

$$ B_{0,0}(t) = 1 $$ $$ B_{0,1}(t) = 1-t $$ $$ B_{1,1}(t) = t $$ $$ B_{0,2}(t) = (1-t)^2 $$ $$ B_{1,2}(t) = 2(1-t)t $$ $$ B_{2,2}(t) = t^2 $$ $$ B_{0,3}(t) = (1-t)^3 $$ $$ B_{1,3}(t) = 3(1-t)^2t $$ $$ B_{2,3}(t) = 3(1-t)t^2 $$ $$ B_{3,3}(t) = t^3. $$

The Bernstein polynomials have a number of useful properties (Farin 1993). They satisfy

- symmetry $$ B_{i,n}(t)=B_{n-i,n}(1-t), $$
- positivity $$ B_{i,n}(t)>=0 $$
- for 0<=t<=1, normalization $$ \sum_{i=0}^nB_{i,n}(t)=1, $$
- and $B_{i,n}$ with $i!=0,n$ has a single unique local maximum of
$$ i^i n^{-n}(n-i)^{n-i}\binom{n}{i} $$
occurring at $t=i/n$.
The envelope $f_n(x)$ of the Bernstein polynomials $B_{i,n}(x)$ for $i=0, 1, ..., n$ (Mabry 2003) is given by $$ f_n(x)=\frac{1}{\sqrt{2\pi nx(1-x)}} $$