Bernstein polynomials,

, are weighted multiples of

and

of the form

, where

is the degree,

is the index running from 0 to

, and

. So, for each degree

, there are

polynomial functions

from

to

. The only zeros of these functions are 0 and 1; the index

counts the multiplicity of the root at 0 and

counts the multiplicity of the root at 1. In addition, they are positive in

, nonnegative in [0,1] and, for each

, they sum to the constant function 1 on

, so they constitute a partition of unity. This is why they can be used to build convex combinations.

In

*Mathematica* these polynomials are denoted as

BernsteinBasis[d,i,s].Thinking of the variable

as a parameter, a Bernstein vector of degree

and parameter

can thus be defined as

, a vector of functions from

to

, with

entries.

On the other hand, if the

roots of unity on the unit circle are

, then for every parameter value

, the linear combinations

are a convex combination of the points

to

. They lie within the convex hull of

and thus within the unit circle.

Regarded as a mapping from

to the plane, these linear combinations

form a curve called the "basic Bézier curve of the points

to

". A component

is called the "Bézier part of

"; the combination

is also called the "Bézier sum of

to

".

Note: "basic" means that we are talking about polynomial curves, not piecewise polynomials!

Snapshot 1: click the buttons one after the other; after any change mouse over the graphics to see explanations

Snapshot 2: the convex components are shown underlined by the corresponding segment chart

Snapshot 3: the convex sum and Bézier's polygon are shown, the length of its components indicated by the segment radius

Snapshot 4: the convex sum and Bézier's polygon are shown, the length of its components indicated by the length sum