1.3.1 Bernstein polynomials

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1.3.1 Bernstein polynomials

The Bernstein polynomials are defined as

$\displaystyle B_{i,n}(t) = \frac{n!}{i!(n-i)!}(1-t)^{n-i}t^i, \;\;\;\;\;\; i=0,\ldots,n\;.$

(1.19)

They form a basis for polynomials (see Sect. 4.4) and have several properties of interest:

Non-negativity: $B_{i,n}(t) \geq 0, \;\;\;\;\;\;\;\;0\leq t \leq 1, \;\;\;i=0,\ldots,n\;$ .
Partition of unity: $\sum_{i=0}^n B_{i,n}(t)= (1-t+t)^n = 1$ (by the binomial theorem).

Symmetry:

$\displaystyle B_{i,n}(t) = B_{n-i,n}(1-t)\;.$

(1.20)

Recursion: $B_{i,n}(t) = (1-t) B_{i,n-1}(t)+ t B_{i-1,n-1}(t)$ with $B_{i,n}(t)=0$ for 0, and $B_{0,0}(t) =1\;$ .

Linear precision:

$\displaystyle t = \sum_{i=0}^n\frac{i}{n}B_{i,n}(t)\;,$

(1.21)

which implies that the monomial can be expressed as the weighted sum of Bernstein polynomials of degree with coefficients evenly spaced in the interval [0,1]. This property is used extensively in Chaps. 4 and 5.

Degree elevation: The basis functions of degree can be expressed in terms of those of degree [106] as:

$\displaystyle B_{i,n}(t) = \left(1- \frac{i}{n+1}\right)B_{i,n+1}(t) + \frac{i+1}{n+1}B_{i+1,n+1}(t)\;,$

(1.22)

where $i = 0,1,\cdots,n$ . Or more generally in terms of basis functions of degree [106] as:

$\displaystyle B_{i,n}(t) = \sum_{j=i}^{i+r}\frac{ \begin{pmatrix}n \cr i \cr\en... ...begin{pmatrix}n+r \cr j \cr\end{pmatrix}}B_{j,n+r}(t),\quad i = 0,1,\cdots,n\;.$

(1.23)

Figure 1.2 shows the Bernstein polynomials of degree 3 and 4. The derivative of a Bernstein polynomial is

$\displaystyle \frac{dB_{i,n}(t)}{dt} = n[B_{i-1,n-1}(t)-B_{i,n-1}(t)]\;,$ (1.24)

where $B_{-1,n-1}(t) = B_{n,n-1}(t) = 0$ .

Figure 1.2: Bernstein polynomials: (a) degree three, (b) degree four
$\begin{figure}\centerline{ \psfig{file=fig/b30.ps,height=3.0in}} \centerline{ (... ...terline{ {\psfig{file=fig/b40.ps,height=3.0in}}} \centerline{ (b)}\end{figure}$

Next: 1.3.2 Arithmetic operations of Up: 1.3 Bézier curves and Previous: 1.3 Bézier curves and Contents Index
December 2009