A Design based on the caustic rays of a Epitrochoid.

Mathematica Notebook for This Page.


From Robert Yates:

Caustics were first introduced and studied by Tschirnhausen in 1682. Other contributors were Huygens, Quetelet, Lagrange, and Cayley.


Caustic is a method of deriving a new curve based on a given curve and a point. A curve derived this way may also be called caustic. Given a curve C and a fixed point S (the light source), catacaustic is the envelope of light rays coming from S and reflected from the curve C. Diacaustic is the envelope of refracted rays. Light rays may also be parallel, as when the light source is at infinity.

The catacaustic of a cardioid (shaped like a apple core in the center)

Caustic do not always generates a curve. For example, the light rays reflected from a parabola's focus do not intersect, therefore its envelope do not form any curve. Another example is illustrated by the catacaustic of a astroid.

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Catacaustic and Diacaustic with parallel rays

Catacaustic of a curve C with parallel rays from one direction generate a curve that is also the diacaustic of the curve C with parallel rays from the opposite direction.

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Catacaustic and diacaustic of sine curve.
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Catacaustic and diacaustic of a ellipse.

Curve relations by caustics

Base Curve Light Source Catacaustic
circle on curve cardioid
circle not on curve limacon of Pascal
circle Infinity nephroid
parabola rays perpendicular to directrix Tschirnhausen's cubic
Tschirnhausen's cubic focus semicubic parabola
cissoid of Diocles focus cardioid
cardioid cusp nephroid
quadrifolium center astroid
deltoid Infinity astroid
equiangular spiral center equiangular spiral
cycloid rays perpendicular to line through cusps cycloid 1/2
y==E^x rays perpendicular y-axis catenary


caustic milk
A photo showing a cardioid formed by light rays reflected in a cup of milk.

More photos: apple juice in glass; 2; crystal shotglass.

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

Caustic (mathematics).

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Plane Curves


  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


  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


  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