# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

Plot a ruled surface

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

ResourceFunction["RuledSurfacePlot"][ plots a ruled surface from the curves | |

ResourceFunction["RuledSurfacePlot"][ plots a ruled surface with |

A ruled surface is generated by the continuous motion of a straight line passing through two curves *c*_{1} and *c*_{2}, and has a parametrization of the form , where *c*_{1} is the directrix curve and *c*_{2} is called the director curve or the ruling.

ResourceFunction["RuledSurfacePlot"] has the same options as ParametricPlot3D.

ResourceFunction["RuledSurfacePlot"][*c*_{1},*c*_{2},*u*] is equivalent to ResourceFunction["RuledSurfacePlot"][*c*_{1},*c*_{2},{*u*,-*π*,*π*},{*v*,-*π*,*π*}].

A simple ruled surface:

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This Manipulate shows how the surface is generated, the right line moves around intersecting the two curves, for definite values of *u* and *v*:

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Modify the default ranges:

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Modify the options for enhanced viewing:

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Plot the generalized hyperbolic paraboloid:

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Define the Plücker conoid:

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Plot the conoid:

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The Möbius strip as a ruled surface:

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A representation of the director curve *c*_{2}(*u*):

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An elliptical hyperboloid is doubly ruled because it can be parametrized in two ways:

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The "minus" chart could easily have been defined in terms of the "plus" one:

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Plot both cases:

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The Gaussian curvature of a ruled surface is everywhere nonpositive:

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This plot exhibits its minima, corresponding to regions which are especially distorted:

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Generalized cylinders and cones have the form of a ruled surface:

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A figure-eight curve:

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Parametrizations of a generalized cylinder and cone using a figure-eight curve:

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Plot the surfaces:

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If the Gaussian curvature of a ruled surface is everywhere zero, then it is said to be a flat surface:

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The tangent developable of a space curve *α* is a ruled surface, whose director curve is the unit tangent vector field to * α*:

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- 1.0.0 – 26 July 2021

This work is licensed under a Creative Commons Attribution 4.0 International License