# Bernstein Polynomials and Convex Bézier Sums

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This Demonstration throws light on the fact that the points that form a basic Bézier curve are convex combinations of the Bézier points to : , where the coefficients are just the Bernstein polynomials of degree and is a parameter running from 0 to 1.

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Contributed by: Ludwig Weingarten (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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

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