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# Channel surface

A channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve. One sheet of the focal surface of a channel surface will be the generating curve.

If the sphere centers lie on a straight line, the channel surface is a surface of revolution. Dupin cyclides form a special class of surfaces which are channel surfaces in two distinct ways: for cyclides both sheets of the focal surface are curves; in fact they are both conic sections.

A section of a torus, a special case of a cyclide. The black lines are the two sheets of the focal surface, which here both degenerate to curves. The surface can be generated as envelopes of spheres centered on these lines.

References

Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed. ed.). Chelsea. p. 219. ISBN 0-8284-1087-9.

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