Wikipedia / Channel surface
Wikipedia / Channel surface
In geometry and topology, a channel surface 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:
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
,
intersect in a curve that fulfills the equations
.
. The last equation is the reason for the following definition.
being at least twice continuously differentiable). The surface defined by the two equations
is the envelope of the given pencil of surfaces.
Canal surface
be a regular space. 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 its directrix. If the radii are constant, it is called a pipe surface.
Parametric representation of a canal surface
The envelope condition
. Hence
form an orthonormal basis, is a parametric representation of the canal surface.
one gets the parametric representation of a pipe surface:
Examples
a) The first picture shows a canal surface with
as directrix and
.
is the following:
.
, i. e. the canal surface is a pipe surface.
.
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).