canal surface

In geometry and topology, 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 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
:\Phi_c: f(,c)=0 , c\in _1,c_2/math>,
two neighboring surfaces $\Phi_c$ and
$\Phi_$ intersect in a curve that fulfills the equations
:$f(,c)=0$ and $f(,c+\Delta c)=0$.

For the limit $\Delta c \to 0$ one gets
$f_c(,c)= \lim_ \frac=0$.
The last equation is the reason for the following definition.
* Let \Phi_c: f(,c)=0 , c\in _1,c_2/math> be a 1-parameter pencil of regular implicit $C^2$ surfaces ($f$ being at least twice continuously differentiable). The surface defined by the two equations
*:$f(,c)=0, \quad f_c(,c)=0$
is the envelope of the given pencil of surfaces.

Canal surface

Let $\Gamma: =(u)=(a(u),b(u),c(u))^\top$ be a regular space curve and $r(t)$ a $C^1$-function with $r>0$ and $, \dot, <\, \dot\,$. 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
:$f(;u):= \big\, -(u)\big\, ^2-r^2(u)=0$
is called a canal surface and $\Gamma$ its directrix. If the radii are constant, it is called a pipe surface.

Parametric representation of a canal surface

The envelope condition
:$f_u(,u)= 2\Big(-\big(-(u)\big)^\top\dot(u)-r(u)\dot(u)\Big)=0$
of the canal surface above is for any value of $u$ the equation of a plane, which is orthogonal to the tangent
$\dot(u)$ 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 $u$) has the distance
d:=\frac (see condition above)
from the center of the corresponding sphere and its radius is $\sqrt$. Hence
:*$=(u,v):= (u)-\frac\dot(u) +r(u)\sqrt \big(_1(u)\cos(v)+ _2(u)\sin(v)\big),$
where the vectors $_1,_2$ and the tangent vector $\dot/\, \dot\,$ form an orthonormal basis, is a parametric representation of the canal surface.''Geometry and Algorithms for COMPUTER AIDED DESIGN''
p. 117

For $\dot=0$ one gets the parametric representation of a pipe surface:
:* $=(u,v):= (u)+r\big(_1(u)\cos(v)+ _2(u)\sin(v)\big).$

Examples

:a) The first picture shows a canal surface with
:#the helix (\cos(u),\sin(u), 0.25u), u\in ,4/math> as directrix and
:#the radius function $r(u):= 0.2+0.8u/2\pi$.
:#The choice for $_1,_2$ is the following:
::$_1:=(\dot,-\dot,0)/\, \cdots\, ,\ _2:= (_1\times \dot)/\, \cdots\,$.
:b) For the second picture the radius is constant:$r(u):= 0.2$, i. e. the canal surface is a pipe surface.
:c) For the 3. picture the pipe surface b) has parameter u\in ,7.5/math>.
: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

*{{cite book
, author1= Hilbert, David
, author-link= David Hilbert
, author2=Cohn-Vossen, Stephan
, title = Geometry and the Imagination
, url= https://archive.org/details/geometryimaginat00davi_0, url-access= registration, edition = 2nd
, year = 1952
, publisher = Chelsea
, page
219
, isbn = 0-8284-1087-9

External links

M. Peternell and H. Pottmann: ''Computing Rational Parametrizations of Canal Surfaces''

Surfaces