Conic Section as Bézier Curve

ω

1

1.

control polygon

Any conic section can be represented as a rational Bézier curve of degree two defined by

C(t)=

B

0,2

(t)

ω

0

P

0

+

B

1,2

(t)

ω

1

P

1

+

B

2,2

(t)

ω

2

P

2

B

0,2

(t)

ω

0

+

B

1,2

(t)

ω

1

+

B

2,2

(t)

ω

2

, where

B

i,n

(t)

are the Bernstein polynomials and

P

i

the control points. It is always possible to write the expression in a standard form such that

ω

0

=

ω

2

=1

. From such a form it is easy to determine the type of the conic section: if

ω

1

>1

, it is a hyperbola; if

ω

1

=1

, it is a parabola; and if

ω

1

<1

, it is an ellipse.