Mathematica Notebook for This Page.
above: A Design based on the caustic rays of a Epitrochoid.
Mathematica Notebook for This Page.
Caustics were first introduced and studied by Tschirnhausen↗ in 1682. Other contributors were Huygens↗, Quetelet↗, Lagrange↗, and Cayley↗. [verbatim from Robert C Yates. 1952]
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.
above: 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.
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.
Catacaustic and diacaustic of sinusoid.
Catacaustic and diacaustic of a ellipse.
| 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 |
above: A photo showing a cardioid formed by light rays reflected in a cup of milk.
More photos: apple juice in glass; 2, crystal shot glass.
See: Websites on Plane Curves, Printed References On Plane Curves.
Robert Yates: Curves and Their Properties.
Wikipedia: Caustic (mathematics)↗.
