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canal surface: directrix is a helix, with its generating spheres
pipe surface: directrix is a helix, with generating spheres
pipe surface: directrix is a helix

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.

  • In technical area canal surfaces can be used for blending surfaces smoothly.

Contents

Envelope of a pencil of implicit surfacesEdit

Given the pencil of implicit surfaces

 .

Two neighboring surfaces   and   intersect in a curve that fulfills the equations

  and  .

For the limit   one gets  . The last equation is the reason for the following definition

  • Let be   a 1-parameter pencil of regular implicit   - surfaces (  is at least twice continuously differentiable). The surface defined by the two equations
     

is the envelope of the given pencil of surfaces.[1]

Canal surfaceEdit

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 surfaceEdit

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 (for parameter  ) has the distance   (s. condition above) 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.[2]

For   one gets the parametric representation of a pipe surface:

  •  
 
pipe knot
 
canal surface: Dupin cyclide

ExamplesEdit

a) The first picture shows a canal surface with
  1. the helix   as directrix and
  2. the radius function  .
  3. The choice for   is the following:
 .
b) For the second picture the radius is constant: , i. e. the canal surface is a pipe surface.
c) For the 3. picture the pipe surface b) has parameter  .
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).

ReferencesEdit

  • Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed.). Chelsea. p. 219. ISBN 0-8284-1087-9.

External linksEdit