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