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

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

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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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Choose two functions:

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Plot the surface:

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Enrique Zeleny

## Version History

• 1.0.0 – 22 June 2020