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