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
![[Snapshot]](HTMLImages/index.en/thumbnail_1.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_2.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_3.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_4.gif)
Embed Interactive Demonstration 
Browse all topics
