Channel surface
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.
Given the pencil of implicit surfaces two neighboring surfaces
intersect in a curve that fulfills the equations For the limit
lim
The last equation is the reason for the following definition.
is the envelope of the given pencil of surfaces.
be a regular space curve 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.
The envelope condition of the canal surface above is for any value of
the equation of a plane, which is orthogonal to the tangent
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.
one gets the parametric representation of a pipe surface:
