Focus and Directrix in a Quadratic Bézier Curve

ω

1.

t

0.5

control polygon

directrix and focus

Any quadratic Bézier curve (with unit parameter

ω=1

) represents a parabolic segment. This Demonstration illustrates the relationship between the disposition of the points and the vertex, locus, and directrix of the corresponding parabola.

You can drag the points

P

0

,

P

1

, and

P

2

. The median of the triangle

Δ

P

0

P

1

P

2

corresponding to the control point

P

1

is perpendicular to the directrix of the parabola, but the vertex and focus are generally not on this line.

The point of maximal curvature in a quadratic Bézier curve is naturally the vertex of the parabola.

You can vary the parameter

ω

of the point

P

1

, providing a rational quadratic Bézier curve, as in the Demonstration "Conic Section as Bézier Curve". See the details of that Demonstration for more information about rational Bézier curves.

A weight

ω<1

or

ω>1

produces an ellipse and a hyperbola, respectively.