Orthoptic and Isoptic


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.




The orthoptic of a astroid with respect to its center is r^2 == (1/2)*Cos[2*θ]^2. [Robert C Yates.]

orthoptic curve

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, Printed References On Plane Curves.

Robert Yates: Curves and Their Properties.