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:
Then find (x,y) such that
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.
- Jan Wassenaar's Curves
- "Courbe Isoptique" at Encyclopédie des Formes Mathématiques Remarquables (in French)
- "Courbe Orthoptique" at Encyclopédie des Formes Mathématiques Remarquables (in French)
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