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Orthoptics of a circle, of some ellipses and hyperbolas

In the geometry of curves, an isoptic is the set of points for which two tangents of a given curve meet at a given angle. The orthoptic is the isoptic whose given angle is a right angle.

Without an invertible Gauss map, an explicit general form is impossible because of the difficulty knowing which points on the given curve pair up.


Orthoptic of a parabola

Take as given the parabola (t,t²) and angle 90°. Find, first, τ such that the tangents at t and τ are orthogonal:

(1,2t)\cdot(1,2\tau)=0 \,
\tau=-1/4t \,

Then find (x,y) such that

(x-t)2t=(y-t^2) \, and (x-\tau)2\tau=(y-\tau^2) \,
2tx-y=t^2 \, and 8t x+16t^2y=-1 \,
x=(4t^2-1)/8t \, and y=-1/4 \,

so the orthoptic of a parabola is its directrix.

The orthoptic of an ellipse is the director circle.


  • J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 58–59. ISBN 0-486-60288-5. 

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