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Spli

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 $$f_n(x)=\frac{1}{\sqrt{2\pi nx(1-x)}}$$