# Focus and Directrix in a Quadratic Bézier Curve

Focus and Directrix in a Quadratic Bézier Curve

Any quadratic Bézier curve (with unit parameter ) 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.

ω=1

You can drag the points , , and . The median of the triangle corresponding to the control point is perpendicular to the directrix of the parabola, but the vertex and focus are generally not on this line.

P

0

P

1

P

2

Δ

P

0

P

1

P

2

P

1

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

You can vary the parameter of the point , 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.

ω

P

1

A weight or produces an ellipse and a hyperbola, respectively.

ω<1

ω>1