Channel surface

canalsurface: 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.

    Envelope of a pencil of implicit surfaces

    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.

    Canal surface

    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 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 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 a orthonormal basis, is a parametric representation of the canal surface.

    For one gets the parametric representation of a pipe surface:

    pipe knot
    canal surface: Dupin cyclide

    Examples

    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).

    References

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

    External links

    • M. Peternell: Computing Rational Parametrizations of Canal Surfaces
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