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.]

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, Plane Curves Books.

Robert Yates: Curves and Their Properties.

2006-05

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Plane Curves

Ancient

  1. Conic Sections
  2. Parabola
  3. Hyperbola
  4. Ellipse
  5. Cissoid
  6. Conchoid
  7. Quadratrix
  8. Archimedean Spiral
  9. Equiangular Spiral
  10. Lituus
  11. Cornu Spiral

Cyclodal

  1. Epitrochoid
  2. Hypotrochoid
  3. Epicycloid and Hypocycloid
  4. Rose Curve
  5. Astroid
  6. Deltoid
  7. Nephroid
  8. Cardioid
  9. Trochoid
  10. Cycloid

Calculus Era

  1. Cassinian Oval
  2. Cross Curve
  3. Folium of Descartes
  4. Piriform
  5. Semicubic Parabola
  6. Tractrix
  7. Trisectrix
  8. Trisectrix of Maclaurin
  9. Lemniscate of Bernoulli
  10. Lemniscate of Gerono
  11. Limacon Of Pascal
  12. Witch of Agnesi
  13. Sine Curve
  14. Catenary
  15. Bezier Curve

Methods

  1. Caustics
  2. Cissoid
  3. Conchoid
  4. Envelope
  5. Evolute
  6. Involute
  7. Geometric Inversion
  8. Orthoptic
  9. Parallel Curve
  10. Pedal Curve
  11. Radial Curve
  12. Roulette

Math of Curves

  1. Geometry: Coordinate Systems for Plane Curves
  2. Coordinate Transformation
  3. Vectors
  4. Naming and Classification of Curves
  1. Cusp
  2. Curvature