Canal area
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)
- Rotation surface (channel surface rail curve is a straight line).
Canal areas play in the
- Representative geometry plays an important role, since its outline can be constructed as the envelope of circles in a perpendicular parallel projection. See outline construction .
- Technology plays an important role as transition surfaces between cylinders.
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
- and .
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.
Canal area
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:
Examples
- 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.
- e) The fifth picture shows a Dupin's cyclide (canal surface).
Note: A slope area is created according to the same principle. The generating sets of surfaces are there cones (pouring cones), the tips of which lie on the guide curve.
Individual evidence
- ↑ envelope (English WP)
- ↑ Geometry and Algorithms for COMPUTER AIDED DESIGN , p. 115
- ↑ Geometry and Algorithms for COMPUTER AIDED DESIGN , p. 117
literature
- Hilbert, David ; Cohn-Vossen, Stephan: Geometry and the Imagination , 2nd ed., 1952, Chelsea, p. 219, ISBN 0-8284-1087-9