Editing Bernstein polynomial (section)

=== Bernstein basis polynomials ===

The <math>\ n + 1\ </math> '''Bernstein basis polynomials''' of degree <math>\ n\ </math> are defined as

: <math>\ b_{\nu,n}(x)\ \equiv\ \binom{n}{\nu}\ x^{\nu} \left( 1 - x \right)^{n - \nu}\ , ~~</math> for <math>~~ \nu = 0\ ,\ \ldots\ , n\ ,</math>

where <math>\ \tbinom{n}{\nu}\ </math> is a [[binomial coefficient]].

So, for example, <math>\ b_{2,5}(x)\ =\ \tbinom{5}{2}x^2(1-x)^3\ =\ 10x^2(1-x)^3 ~.</math>

The first few Bernstein basis polynomials for blending {{math|1, 2, 3}} or {{math|4}} values together are:
: <math>
\begin{align}
b_{0,0}(x) & = 1\ , \\
b_{0,1}(x) & = 1 - x\ , & b_{1,1}(x) & = x \\
b_{0,2}(x) & = (1 - x)^2\ , & b_{1,2}(x) & = 2x(1 - x)\ , & b_{2,2}(x) & = x^2 \\
b_{0,3}(x) & = (1 - x)^3\ , & b_{1,3}(x) & = 3x(1 - x)^2\ , & b_{2,3}(x) & = 3x^2(1 - x)\ , & b_{3,3}(x) & = x^3 ~.
\end{align}
</math>
:

The Bernstein basis polynomials of degree <math>\ n\ </math> form a [[basis (linear algebra)|basis]] for the [[vector space]] <math>\ \Pi_n\ </math> of polynomials of degree at most <math>\ n\ ,</math> all with real coefficients.