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 **pipe surface**. Simple examples are:

- right circular cylinder (pipe surface, directrix is a line, the axis of the cylinder)
- torus (pipe surface, directrix is a circle),
- right circular cone (canal surface, directrix is a line (the axis), radii of the spheres not constant),
- surface of revolution (canal surface, directrix is a line),

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

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

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

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:

## Examples

- a) The first picture shows a canal surface with
- the helix as directrix and
- the radius function .
- 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).

## References

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