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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 RadialCurve

Source Notebook

Compute the radial curve of a given curve

Contributed by: Eric W. Weisstein

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

computes the radial curve of the curve c in parameter t with respect to the point p.

Details and Options

The radial curve is the locus traced by the curvature vector of the given curve, starting at a given point in the plane.

Examples

Basic Examples (3) 

Obtain the parametric form of an astroid:

In[1]:=
astroid = Entity["PlaneCurve", "Astroid"]["ParametricEquations"][1][t]
Out[1]=

Compute the parametric form for the radial curve of the astroid:

In[2]:=
rcastroid = ResourceFunction["RadialCurve"][astroid, {0, 0}, t] // FullSimplify
Out[2]=

Plot the radial curve of the astroid (red):

In[3]:=
ParametricPlot[Evaluate[{rcastroid, astroid}], {t, 0, 2 \[Pi]}, Ticks -> None, PlotStyle -> {Red, {}}, Epilog -> {{Red, PointSize[.03], Point[{0, 0}]}}]
Out[3]=

Scope (2) 

Define an epicycloid:

In[4]:=
ec = Entity["PlaneCurve", "Epicycloid"]["ParametricEquations"][m, n][ t]
Out[4]=

A table of plots of epicycloids for different values of m, n:

In[5]:=
GraphicsGrid[ Table[Module[{rc = ResourceFunction["RadialCurve"][ec, {0, 0}, t]}, ParametricPlot[Evaluate[{rc, ec}], {t, .01, 2 \[Pi]}, Ticks -> None, PlotStyle -> {Red, {}}, Epilog -> {{Red, PointSize[.03], Point[{0, 0}]}}]], {n, 4}, {m, 4}]]
Out[5]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 18 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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