# 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.

2006-05

If you have a question, put $5 at patreon and message me.