Isoptic

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.

Example

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.

References

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