Raising the Degree for Bézier Curves

number of iterations k

1

A Bézier curve in the plane is given by parametric equations of the form

C(t)=

n

∑

i=0

B

i,n

(t)

P

i

, where

P

1

,

P

2

,…,

P

n

are points in the plane called control points and

B

i,n

(t)

is the

th

i

Bernstein polynomial of degree

n

. This parametrization can be changed (without changing the curve) via a recursive procedure outlined in the Details section that generates a new set of control points larger by 1 at each iteration. The figure shows the Bézier curve, the original control polygon (obtained by connecting the control points with line segments), and the control polygon determined by the control points after some iterations of the recursive procedure. This artificial process is useful in CAGD to increase the flexibility for the designer. It may also be used in numerous contexts where several Bézier curves need to be represented with the same degree, for example to join two curves by a Bézier surface, or to compare two curves, etc.