Bernstein polynomials: The Bernstein polynomials of degree n are defined
by
B^{n}_{i}(t) = (

n i

) t^{i}(1t)^{ni}, i = 0, 1, 2,
..., n

Here is an easy way to remember them: In order to find the Bernstein
polynomials of degree 3, we simply expand
(t+(1t))^{3} = t^{3} + 3t^{2}(1t) +
3t(1t)^{2} + (1t)^{3}
Now, B^{3}_{3}(t)=t^{3},
B^{3}_{2}(t)=3t^{2}(1t),
B^{3}_{1}(t)=3t(1t)^{2},
B^{3}_{0}(t)=(1t)^{3}
In general, (t+(1t))^{n} =

n
S
i=0

(

n i

)t^{i}(1t)^{ni} =

n
S
i=0

B^{n}_{i}(t)

Propositions of Bernstein Polynomials
 (Recursion) The Bernstein polynomials satisfy the following
recursion formula: B^{n}_{i}(t) =
(1t)B^{n1}_{i}(t) +
tB^{n1}_{i1}(t)
Proof:
B^{n}_{i}(t) = (

n i

) t^{i}(1t)^{ni} =
(

n1 i

) t^{i}(1t)^{ni} +
(

n1 i1

) t^{i}(1t)^{ni}


= (1t)B^{n1}_{i}(t) +
tB^{n1}_{i1}(t)

 (Forming a partition of unity) The set of Bernstein polynomials of
degree n {B^{n}_{i}(t)  t= 0, 1, ..., n} form a
partition of unity over the interval [0,1].
Proof:
B^{n}_{i}(t) = (

n i

) t^{i}(1t)^{ni} >
0 for 0 < t < 1.

=

n
S
i=0

B^{n}_{i}(t) =

n
S
i=0

(

n i

) t^{i}(1t)^{ni} =
[t+(1t)]^{n} = 1

Bernstein polynomials in barycentric coordinates
Example: Consider the line segment I=[0,1] in R^{1}. Using the
barycentric coordinates of the line segment I, every point p in I can be
described by coordinates u=t, v=1t, with t in I. With respect to these
coordinates, the Bernstein polynomial can be written as
B^{n}_{i}(t) = B^{n}_{i,j}(u,v) =

n!
i! j!

u^{i}v^{j}

with i+j=n; u+v=1; u,v,i,j > 0. The
B^{n}_{i,j}(u,v) can thus be thought of as terms in the
binomial expansion of (u+v)^{n}. This principle can be carried
over to triangles in R^{2}which leads to the generalized
Bernstein polynomials.
Generalized Bernstein polynomials: Recall that every point p in the
interior of a triangle with vertices U, V, and W can be expressed in
terms of the barycentric coordinates (u,v,w) which are determined by
p(u,v,w) = uU + vV + wW, subject to the constraints u+v+w=1; u,v,w
> 0. With the help of barycentric coordinates it is now easy to
define the Bernstein polynomials associated with a base triangle. Using
the following relations,
(u+v+w)^{n} =

S
i,j,k>0
i+j+k=n

n!
i! j! k!

u^{i}v^{j}w^{k}

(u+[v+w])^{n} =

n
S
i=0

(

n i

) u^{i}(v+w)^{ni}

(v+w)^{n1} =

n1
S
i=0

(

ni j

) v^{j}w^{nij}

the generalized Bernstein polynomials of degree n are defined as
B^{n}_{i,j,k}(u,v,w) =

n!
i! j! k!

u^{i}v^{j}w^{k}

with i+j+k=n, u+v+w=1, i,j,k,u,v,w > 0
For ease of notation, we define I:=(i,j,k)^{T},
u:=(u,v,w)^{T} and I=i+j+k, u=u+v+w.
Then the generalized Bernstein polynomials reduce to
B^{n}_{I}(u) =

n!
i! j! k!

u^{i}v^{j}w^{k} with u=1.

On the edges of the base triangle, the generalized Bernstein polynomials
reduce to the ordinary Bernstein polynomials. For example,
B^{n}_{0,j,k}(0,v,w) = B^{n}_{j}(v) =
B^{n}_{k}(w). Clearly, every generalized Bernstein
polynomial B^{n}_{I} can be interpreted as a
(polynomial) parametric formula for a surface patch (function) over the
base triangle. The maximum of this function occurs at
u=I/n.
Propositions:
 By their definition, it is clear that the Bernstein polynomials
satisfy the normalization condition
where the sum is taken over all possible vectors I which satisfy
the conditions I= n and I > 0. This involves a total
of
terms.
 Since the generalized Bernstein polynomials are linearly independent,
they form a basis for an (n+1)(n+2)/2 dimensional linear subspace of the
space of polynomials of degree (n, n). Every element X in the linear
subspace spanned by the B^{n}_{I} has a unique
expansion
X(u)=

S
I=n

b_{I}B^{n}_{I}(u)

Using the Bernstein polynomials as a basis function, we can define a
Bézier curve or Bézier polynomial of degree n (in terms of
Bernstein polynomials) to be a curve in parametric form
X(t) =

n
S
i=0

b_{i}B^{n}_{i}(t)

with coefficients b_{i} in R^{d}, d=1, 2, 3. In general
we take the b_{i} to be vectors in R^{2} or
R^{3}, and refer to them as Bézier points. In the case
where the b_{i} are real numbers, we call them Bézier
ordinates. The polygon formed by connecting the Bézier points is
called the Bézier polygon.