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.
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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.
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A weight or produces an ellipse and a hyperbola, respectively.
ω<1
ω>1