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.



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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