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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 CatacausticCurve

Source Notebook

Compute the catacaustic of a curve

Contributed by: Eric W. Weisstein

ResourceFunction["CatacausticCurve"][c,{a,b},t]

gives an equation for the catacaustic of a curve c specified parametrically with respect to a point {a,b} in terms of the variable t.

A catacaustic is a curve that is the envelope of rays emanating from a specified point (or a point at infinite distance producing parallel rays) for a given mirror shape. The point from which the rays emanate is called the radiant point.
The catacaustic is an evolute of the orthotomic.

Examples

Basic Examples (2) 

The catacaustic for a Tschirnhausen cubic:

In[1]:=
FullSimplify[ ResourceFunction[ "CatacausticCurve"][{3 (t^2 - 3), t (t^2 - 3)}, {x, y}, t]]
Out[1]=

Plot the catacaustic:

In[2]:=
ResourceFunction["CatacausticCurvePlot"][{3 (t^2 - 3), t (t^2 - 3)}, {-8, 0}, {t, -3, 3, .2}, Axes -> True, PlotRange -> {{-10, 20}, {-18, 18}}]
Out[2]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 10 March 2020

Source Metadata

  • Citation:
    • Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.

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