Make Your Own Ruled Surface

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A ruled surface is a surface that has a parametrization of the form , where is the directrix and the director curve or ruling. Such a surface can be generated by the continuous motion of a straight line in space. Here you can choose to be a constant multiple of one of the functions , , and ( is the constant) and to be a constant multiple of one of the functions , , and ( is the constant). Ruled surfaces occur in architecture.

Contributed by: Enrique Zeleny (August 2014)
Open content licensed under CC BY-NC-SA



The snapshots show some well-known ruled surfaces.

Snapshot 1: right conoid

Snapshot 2: Plücker's conoid

Snapshot 3: helicoid


[1] W. L. Edge, The Theory of Ruled Surfaces, Cambridge: Cambridge University Press, 1931.

[2] O. Knill. "Math21a Fall 2009, Gallery of Ruled Surfaces." (2009)

[3] A. Gray, "Ruled Surfaces," Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., Boca Raton, FL: CRC Press, 1993 pp. 431–456.

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