Channel surface

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canal surface: directrix is a helix, with its generating spheres
pipe surface: directrix is a helix, with generating spheres
pipe surface: directrix is a helix

In geometry and topology, a channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its directrix. If the radii of the generating spheres are constant, the canal surface is called a pipe surface. Simple examples are:

Canal surfaces play an essential role in descriptive geometry, because in case of an orthographic projection its contour curve can be drawn as the envelope of circles.

  • In technical area canal surfaces can be used for blending surfaces smoothly.

Envelope of a pencil of implicit surfaces[edit]

Given the pencil of implicit surfaces


two neighboring surfaces and intersect in a curve that fulfills the equations

and .

For the limit one gets . The last equation is the reason for the following definition.

  • Let be a 1-parameter pencil of regular implicit surfaces ( being at least twice continuously differentiable). The surface defined by the two equations

is the envelope of the given pencil of surfaces.[1]

Canal surface[edit]

Let be a regular space curve and a -function with and . The last condition means that the curvature of the curve is less than that of the corresponding sphere. The envelope of the 1-parameter pencil of spheres

is called a canal surface and its directrix. If the radii are constant, it is called a pipe surface.

Parametric representation of a canal surface[edit]

The envelope condition

of the canal surface above is for any value of the equation of a plane, which is orthogonal to the tangent of the directrix. Hence the envelope is a collection of circles. This property is the key for a parametric representation of the canal surface. The center of the circle (for parameter ) has the distance (see condition above) from the center of the corresponding sphere and its radius is . Hence

where the vectors and the tangent vector form an orthonormal basis, is a parametric representation of the canal surface.[2]

For one gets the parametric representation of a pipe surface:

pipe knot
canal surface: Dupin cyclide


a) The first picture shows a canal surface with
  1. the helix as directrix and
  2. the radius function .
  3. The choice for is the following:
b) For the second picture the radius is constant:, i. e. the canal surface is a pipe surface.
c) For the 3. picture the pipe surface b) has parameter .
d) The 4. picture shows a pipe knot. Its directrix is a curve on a torus
e) The 5. picture shows a Dupin cyclide (canal surface).


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

External links[edit]