Channel surface: Leitkurve is a helix , with generating spheres
Pipe surface: Leitkurve is a helix, with generating spheres
Pipe surface: Guide curve is a helix
A channel surface is the enveloping surface of a set of spheres , the centers of which lie on a given curve, the guide curve or directrix . If the radii of the sphere are constant, the channel surface is called a pipe surface . Simple examples are
Circular cylinder (pipe surface, guide curve is a straight line (cylinder axis), spherical radii are constant)
Torus (pipe surface, guide curve is a circle, spherical radii are constant)
Circular cone (channel surface, guide curve is a straight line (cone axis), spherical radii are not constant)
Enveloping surface of an implicit family of surfaces
The array of areas is given
The intersection curve of two adjacent surfaces and
satisfies the equations
For the border crossing results
. The last equation is the reason for the following definition
Let it be a 1-parameter family of regular implicit - surfaces ( is at least 2-times continuously differentiable).
The by the two equations
defined area is called the envelope of the given set of areas.
Let it be a regular space curve and a function with and . The last condition means that the curve is less curved than the corresponding sphere.
The envelope of the one-parameter set of spheres
is called the channel surface and its guide curve or direct rix . If the radius function is constant, the channel area is called the pipe area .
Parameter representation of a duct area
The envelope condition of the above channel area
is a plane equation for a plane perpendicular to the tangent of
the guide curve for each parameter value . So the envelope is the union of circles. This observation is the key to a parametric representation of the canal area. The center of the circle (for a parameter ) has the distance
(see condition above) from the center of the sphere and the radius .
where the vectors together with the tangent vector form an orthonormal basis, is a parametric representation of the channel area.
The parameter representation of a pipe area results for:
Canal surface: Dupin's cyclids
a) The first picture (from above) shows a canal area with
the helix (helix) as the guide curve and
the radius function .
The choice for is:
b) In the second picture the radius is constant:, d. H. the channel surface is a pipe surface.
c) In the third picture the pipe area from b) has parameters .
d) The fourth picture shows a pipe knot. The guide curve runs on a torus.