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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 CatacausticCurvePlot

Source Notebook

Plot the catacaustic of a curve

Contributed by: Eric W. Weisstein

ResourceFunction["CatacausticCurvePlot"][c,{a,b},{t,t0,t1,dt}]

plots the catacaustic of a curve c specified parametrically with respect to a point {a,b} in terms of the variable t, in the range from t0 to t1 with rays spaced in steps of dt.

Details and Options

Rays forming the catacaustic are drawn with spacing dt in terms of the curve parameter t. The dt step size affects only which rays are plotted but doesn't impact the resulting catacaustic.
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 (1) 

The catacaustic for a Tschirnhausen cubic:

In[1]:=
ResourceFunction[ "CatacausticCurvePlot"][{3 (t^2 - 3), t (t^2 - 3)}, {-8, 0}, {t, -3, 3, .2}, Axes -> True, PlotRange -> {{-10, 20}, {-18, 18}}]
Out[1]=

Scope (1) 

Catacaustics for a circle:

In[2]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/56445331-1942-4928-8f8e-3aa7d04b6629"]
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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