Channel surface

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.

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.

References

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