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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 TangentDevelopableSurface

Source Notebook

Compute the tangent developable surface of a curve

Contributed by: Alfred Gray

ResourceFunction["TangentDevelopableSurface"][c,t,{u,v}]

gives the developable surface generated by the parametrized curve c[t] and its tangents in variables u,v.

Details and Options

The tangent developable surface of a curve is generated by the tangent vector field of the curve.

Examples

Basic Examples (2) 

The tangent developable surface of a helix:

In[1]:=
helix = Entity["SpaceCurve", "Helix"]["ParametricEquations"][1, 2][t]
Out[1]=
In[2]:=
tdhelix = ResourceFunction["TangentDevelopableSurface"][helix, t, {u, v}] // Simplify
Out[2]=
In[3]:=
ParametricPlot3D[Evaluate[tdhelix], {u, 0, 12}, {v, 0, 2 \[Pi]}, PlotPoints -> {40, 15}]
Out[3]=

Using the Viviani’s curve:

In[4]:=
viviani[a_, t_] := {a (1 + Cos[t]), a Sin[t], 2 a Sin[t/2]}
In[5]:=
tdviviani = ResourceFunction["TangentDevelopableSurface"][viviani[1, t], t, {u, v}] // Simplify
Out[5]=
In[6]:=
ParametricPlot3D[Evaluate[tdviviani], {u, 0, 4 \[Pi]}, {v, -4, 4}, PlotPoints -> {40, 15}]
Out[6]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 03 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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