Orthoptic and Isoptic
Description
A orthoptic of a curve is the locus of all points where the curve's tangents meet at right angles. If the given angle is other than π/2, it is called isoptic.
History
Formula
Properties
The orthoptic of a astroid with respect to its center is r^2 == (1/2)*Cos[2*θ]^2. [Robert C Yates.]

curves relation by Isoptics
Base Curve | Angle | Isoptic |
---|---|---|
parabola | ? | hyperbola |
parabola | π/2 | directrix |
cardioid | π/2 | circle, limacon of Pascal? |
deltoid | π/2 | inscribed circle |
astroid | π/2 | quadrifolium |
equiangular spiral | π/2 | same? equiangular spiral |
epicycloid | ? | epitrochoid |
hypocycloid | ? | hypotrochoid |
sinusoidal spiral | any? | sinusoidal spiral |
cycloid | ? | curtate or prolate cycloid |
Related Web Sites
See: Websites on Plane Curves, Plane Curves Books .
Robert Yates: Curves and Their Properties .