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 Orthotomic (1.0.0) current version: 1.0.1 »

Source Notebook

Compute the orthotomic of a curve

Contributed by: Eric W. Weisstein  |  Enrique Zeleny

ResourceFunction["Orthotomic"][c,t]

computes the orthotomic in parameter t of a curve c with respect to the point {0,0}.

ResourceFunction["Orthotomic"][c,p,t]

computes the orthotomic with respect to the point p.

Details and Options

The orthotomic curve (also known as secondary caustic) is the reflection of the rays of a source in the tangents of points of a curve.
The orthotomic of a curve is the homothetic image (magnified by a factor of 2 with respect to the center of similarity p) of the pedal.

Examples

Basic Examples (2) 

Orthotomic of a circle with respect to the point {1,1}:

In[1]:=
orthcircle = ResourceFunction["Orthotomic"][{Cos[t], Sin[t]}, {1, 1}, t] // Simplify
Out[1]=
In[2]:=
ParametricPlot[ Evaluate[{{Cos[t], Sin[t]}, orthcircle}], {t, 0, 2 \[Pi]}, PlotRange -> 3.4]
Out[2]=

Orthotomic of a figure eight:

In[3]:=
eight[t_] := {Sin[t], Cos[t] Sin[t]}
In[4]:=
Manipulate[ ParametricPlot[ Evaluate[{eight[t], ResourceFunction["Orthotomic"][eight[t], p, t]}], {t, 0, 2 \[Pi]}, PlotRange -> 3.4], {{p, {0, 0}}, {-\[Pi], -\[Pi]}, {\[Pi], \[Pi]}}]
Out[4]=

Properties and Relations (1) 

Pedal curves look similar to the orthotomic:

In[5]:=
pc = ResourceFunction["PedalCurve"][{Cos[t], Sin[t]}, {1, 1}, t] // Simplify
Out[5]=
In[6]:=
ParametricPlot[ Evaluate[{{Cos[t], Sin[t]}, ResourceFunction["Orthotomic"][{Cos[t], Sin[t]}, {1, 1}, t], pc}], {t, 0, 2 \[Pi]}, PlotRange -> 3.4]
Out[6]=

Version History

  • 1.0.1 – 29 March 2022
  • 1.0.0 – 15 April 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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