Function Repository Resource:

TranslationSurface

Compute a translation surface parametrization

Contributed by: Wolfram Staff (original content by Alfred Gray)

ResourceFunction["TranslationSurface"][c₁,c₂,{t,u,v}]

computes a translation surface parametrized by u and v from two curves c₁ and c₂ parametrized by t.

Details and Options

A translation surface is the surface formed by moving the curve c₁ (first generatrix) parallel to itself such that a point of the curve moves along the curve c₂ (second generatrix), giving the surface formed by the parallel translation.

ResourceFunction["TranslationSurface"] computes a translation surface from curves c₁ and c₂ depending on t with a parametrization in terms of variables u and v.

Examples

Basic Examples (7)

A plane is the translation surface of two intersecting lines:

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Plot it:

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A cylinder is the translation surface of a line translated along a circle:

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Plot it:

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A circle translated along a cosine curve:

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Plot it:

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Form elliptic and hyperbolic paraboloids by translating one parabola along another:

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$ParametricPlot3D[ ResourceFunction[ "TranslationSurface"][{t, 0, t^2}, {0, t, #}, {t, u, v}], {u, -2 Pi, 2 Pi}, {v, -2 Pi, 2 Pi}, BoxRatios -> {1, 1, 1}] & /@ {t^2, -t^2}$

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Translate one helix (red) along another helix (blue):

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Show[ParametricPlot3D[
Evaluate[ResourceFunction[
"TranslationSurface"][{Cos[t], Sin[t], t/4}, {Cos[t], Sin[t], t/4}, {t, u, v}]], {u, 0, 2 Pi}, {v, 0, 8 Pi}, MeshFunctions -> {#3 &}, PlotStyle -> Opacity[.5], PlotPoints -> 80], ParametricPlot3D[{{Cos[t], Sin[t], t/4}, {Cos[t] + 1, Sin[t], t/4}}, {t, 0, 8 Pi}, PlotStyle -> {{Thickness[.01], Blue}, {Thickness[.01], Red}}], Graphics3D[{Thickness[.01], Line[{{1, 0, 0}, {1, 0, 8}}]}]]

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Form an egg box surface by translating one sine along another:

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For the Bohemian dome, the two generatrices are circles:

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Properties and Relations (5)

A sample surface:

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For the coefficients of the second fundamental form, the second coefficient f is equal to zero for a translation surface:

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A surface generated by a polynomial translated along a parabola:

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$ParametricPlot3D[ ResourceFunction[ "TranslationSurface"][{t, 0, (t^2 - 1)^2}, {0, t, -t^2}, {t, u, v}], {u, -1.5, 1.5}, {v, -1.5, 1.5}]$

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There is symmetry as when the order of the curves is reversed, the same surface is obtained:

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$ParametricPlot3D[ ResourceFunction["TranslationSurface"][{0, t, -t^2}, s, {t, u, v}], {u, -1.5, 1.5}, {v, -1.5, 1.5}]$

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This surface is equivalent to the surface of revolution of the sinusoid:

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The first Scherk minimal surface:

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Code to get a curve with prescribed curvature (intrinsic curvature):

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$intrinsic[f_, a_, {c_, d_, e_}, {min_, max_}][t_] := Module[{x, y, \[Theta], sol}, eqic = {x'[s] == Cos[\[Theta][s]], y'[s] == Sin[\[Theta][s]], \[Theta]'[s] == f[s], x[a] == c, y[a] == d, \[Theta][a] == e}; sol = NDSolve[eqic, {x, y, \[Theta]}, {s, min, max}]; {x[t], y[t]} /. sol[[1]] ]$

Choose two functions:

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$f = intrinsic[3 Sin[#] &, 0, {0, 0, 0}, {-10 \[Pi], 10 \[Pi]}]; g = intrinsic[3 Cos[#] & , 0, {0, 0, 0}, {-10 \[Pi], 10 \[Pi]}]; ResourceFunction["TranslationSurface"][Insert[f[t], 0, 2], Insert[g[t], 0, 1], {t, u, v}]$