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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 EvolutoidCurve

Source Notebook

Compute the evolutoid of a curve

Contributed by: Jan Mangaldan

ResourceFunction["EvolutoidCurve"][c,θ,t]

computes the evolutoid with angle θ of the curve c defined with respect to the variable t.

Details

An evolutoid is the envelope of all the lines that intersect a plane curve and are at a constant angle θ from the curve's tangents.
When , the evolute is obtained.

Examples

Basic Examples (3) 

Parametric equations for an ellipse:

In[1]:=
ellipse[a_, b_, t_] := {a Cos[t], b Sin[t]}

Compute the evolutoid of an ellipse with angle equal to GoldenAngle:

In[2]:=
ResourceFunction["EvolutoidCurve"][ellipse[Sqrt[2], 1, t], GoldenAngle, t] // Simplify
Out[2]=

Plot the ellipse and its evolutoid:

In[3]:=
\[Alpha] = ellipse[Sqrt[2], 1, t]; \[Epsilon] = ResourceFunction["EvolutoidCurve"][\[Alpha], GoldenAngle, t]; ParametricPlot[Evaluate[{\[Alpha], \[Epsilon]}], {t, 0, 2 \[Pi]}]
Out[3]=

Properties and Relations (2) 

When , EvolutoidCurve is the same as the resource function EvoluteCurve:

In[4]:=
cc[t_] = Entity["PlaneCurve", "Cardioid"]["ParametricEquations"][1][t]
Out[4]=
In[5]:=
ResourceFunction["EvolutoidCurve"][cc[t], \[Pi]/2, t] == ResourceFunction["EvoluteCurve"][cc[t], t] // Simplify
Out[5]=

Demonstrate the effect of changing θ in EvolutoidCurve:

In[6]:=
Manipulate[ ParametricPlot[ Evaluate[{cc[t], ResourceFunction["EvolutoidCurve"][cc[t], \[Theta], t]}], {t, 0, 2 \[Pi]}], {{\[Theta], \[Pi]/2}, 0, \[Pi]}, SaveDefinitions -> True]
Out[6]=

Version History

  • 1.0.0 – 11 February 2021

Source Metadata

  • Citation:
    • Shikin, E. V., Handbook and Atlas of Curves. Boca Raton, FL: CRC Press, 1995.

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