Editing Bernstein polynomial (section)

=== Bernstein polynomials ===

A linear combination of Bernstein basis polynomials

:<math>\ B_n(x)\ \equiv\ \sum_{\nu=0}^{n} \beta_{\nu} b_{\nu,n}(x)\ </math>

is called a '''Bernstein polynomial''' or '''polynomial in Bernstein form''' of degree <math>\ n ~.</math><ref name="Lorentz">{{harvnb|Lorentz|1953}}</ref> The coefficients <math>\ \beta_\nu\ </math> are called '''Bernstein coefficients''' or '''Bézier coefficients'''.

The first few Bernstein basis polynomials from above in [[monomial]] form are:
: <math>
\begin{align}
b_{0,0}(x) & = 1\ , \\
b_{0,1}(x) & = 1 - 1x\ , & b_{1,1}(x) & = 0 + 1x \\
b_{0,2}(x) & = 1 - 2x + 1x^2, & b_{1,2}(x) & = 0 + 2x - 2x^2\ , & b_{2,2}(x) & = 0 + 0x + 1x^2 \\
b_{0,3}(x) & = 1 - 3x + 3x^2 - 1x^3\ , & b_{1,3}(x) & = 0 + 3x - 6x^2 + 3x^3\ , & b_{2,3}(x) & = 0 + 0x + 3x^2 - 3x^3, & b_{3,3}(x) & = 0 + 0x + 0x^2 + 1x^3 ~.
\end{align}
</math>
: