Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 PedalCurve

Source Notebook

Compute the pedal of a curve

Contributed by: Alfred Gray

ResourceFunction["PedalCurve"][c,t]

computes the pedal curve of a curve c with parameter t.

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

computes the pedal curve from the point p with parameter t.

Details and Options

The curve c must be defined in two dimensions.
If no point p is given, the origin {0,0} is used.
The pedal of a given curve c from a fixed point p is the locus of the foot of the perpendicular from p to the tangent to c.

Examples

Basic Examples (3) 

Define a cardioid:

In[1]:=
cardiod = Entity["PlaneCurve", "Cardioid"][ EntityProperty["PlaneCurve", "ParametricEquations"]][1]
Out[1]=

Compute the pedal curve of the cardioid:

In[2]:=
pc = ResourceFunction["PedalCurve"][cardiod[t], t] // Simplify
Out[2]=

Plot the cardioid (blue) along with its pedal curve (orange):

In[3]:=
ParametricPlot[{cardiod[t], pc}, {t, 0, 2 \[Pi]}]
Out[3]=

Scope (2) 

The pedal curve of an ellipse from different points:

In[4]:=
GraphicsGrid[ Partition[ Table[ParametricPlot[ Evaluate[{{Sin[t], .5 Cos[t]}, ResourceFunction["PedalCurve"][{Sin[t], .5 Cos[t]}, p, t]}], {t, 0, 2 \[Pi]}, Epilog -> Point[p], Axes -> False], {p, {{.5, 0}, {1, 0}, {0, .25}, {0, 1}}}], 2]]
Out[4]=

Nested pedal curves:

In[5]:=
ParametricPlot[ Evaluate[ NestList[ TrigFactor[ResourceFunction["PedalCurve"][#, t]] &, {t, Cos[t]}, 5]], {t, -2 \[Pi], 2 \[Pi]}]
Out[5]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 20 February 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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