# Bernstein Polynomials and Convex Bézier Sums

Bernstein Polynomials and Convex Bézier Sums

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

ℬ

d

p

0

p

d

ℬ

d

d

∑

i=0

B

i,d

p

i

B

i,d

d+1

d

s

This curve lies entirely within the convex hull of its Bézier points and so in this Demonstration is inside the unit circle. (Recall that the convex hull of the points to consists of all convex combinations , where each is restricted to belong to the interval and the have to sum to unity: =1.)

p

0

p

d

d

∑

i=0

γ

i

p

i

γ

i

[0,1]

γ

i

d

∑

i=0

γ

i

The first control sets the number of Bézier points, which are chosen here as the roots of unity on the unit circle and count from to , with being the degree of the Bernstein polynomials belonging to the convex Bézier sum.

n

n

p

0

p

d

d=n-1

n

Click the "curve" button to see the curve (s). The curve for the whole parameter range is thin while the curve from 0 up to the current parameter value of is thick.

ℬ

d

s

Click the "points" button to see the vectors from the origin to the Bézier points.

n

Click the "convex sum" button to see the convex combination (s), which is built up (starting with the origin) by putting each component (s) as a vector on top of its predecessor (s), and thus creating "Bézier's polygon".

d

∑

i=0

B

i,d

p

i

B

i,d

p

i

B

i-1,d

p

i-1

Click the "convex components" button to see the individual component (s) as a multiple of its Bézier point vector , which can be seen after pressing the button; to emphasize this, click the "segment chart" button.

B

i,d

p

i

p

i

Click the button "length sum" to see that the Bernstein factors (s) indeed form a partition of unity of the form (s)=1, which covers 1 as the vector ranging from the origin (0,0) to the point in the plane.

B

i,d

d

∑

i=0

B

i,d

(1,0)

The response time shortens if you use only some of the buttons.

To each number of points there corresponds a different Bézier curve. Choose a number!

n

Almost every item in the graphic is annotated, so mouseover them to see the explanations.