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, its directrix. If the radii of the generating spheres are constant the canal surface is called 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.
Given the pencil of implicit surfaces
Two neighboring surfaces and
intersect in a curve that fulfills the equations
For the limit one gets
The last equation is the reason for the following definition
is the envelope of the given pencil of surfaces.

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 canal surface and its directrix. If the radii are constant, it is called 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 has the distance

from the center of the corresponding sphere and its radius is. Hence
where the vectors and the tangenten vector form an orthonormal basis, is a parametric representation of the canal surface.
For one gets the parametric representation of a pipe surface:

Examples