{{short description|Surface formed from spheres centered along a curve}} [[File:Canal-helix-s.svg|400px|thumb|canal surface: directrix is a helix, with its generating spheres]] 400px|thumb|pipe surface: directrix is a helix, with generating spheres 300px|thumb|pipe surface: directrix is a helix
In geometry and topology, a '''channel surface''' 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 :<math>\Phi_c: f({\mathbf x},c)=0 , c\in [c_1,c_2]</math>, two neighboring surfaces <math>\Phi_c</math> and <math>\Phi_{c+\Delta c}</math> intersect in a curve that fulfills the equations :<math> f({\mathbf x},c)=0</math> and <math>f({\mathbf x},c+\Delta c)=0</math>.
For the limit <math>\Delta c \to 0</math> one gets <math>f_c({\mathbf x},c)= \lim_{\Delta c \to \ 0} \frac{f({\mathbf x},c)-f({\mathbf x},c+\Delta c)}{\Delta c}=0</math>. The last equation is the reason for the following definition. * Let <math>\Phi_c: f({\mathbf x},c)=0 , c\in [c_1,c_2]</math> be a 1-parameter pencil of regular implicit <math>C^2</math> surfaces (<math>f</math> being at least twice continuously differentiable). The surface defined by the two equations *:<math> f({\mathbf x},c)=0, \quad f_c({\mathbf x},c)=0 </math> is the '''envelope''' of the given pencil of surfaces.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf ''Geometry and Algorithms for COMPUTER AIDED DESIGN''], p. 115</ref>
== Canal surface == Let <math>\Gamma: {\mathbf x}={\mathbf c}(u)=(a(u),b(u),c(u))^\top</math> be a regular space curve and <math>r(t)</math> a <math>C^1</math>-function with <math>r>0</math> and <math>|\dot{r}|<\|\dot{\mathbf c}\|</math>. 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 :<math>f({\mathbf x};u):= \big\|{\mathbf x}-{\mathbf c}(u)\big\|^2-r^2(u)=0</math> is called a '''canal surface''' and <math>\Gamma</math> its '''directrix'''. If the radii are constant, it is called a '''pipe surface'''.
== Parametric representation of a canal surface == The envelope condition :<math>f_u({\mathbf x},u)= 2\Big(-\big({\mathbf x}-{\mathbf c}(u)\big)^\top\dot{\mathbf c}(u)-r(u)\dot{r}(u)\Big)=0</math> of the canal surface above is for any value of <math>u</math> the equation of a plane, which is orthogonal to the tangent <math>\dot{\mathbf c}(u)</math> 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 <math>u</math>) has the distance <math>d:=\frac{r\dot{r}}{\|\dot{\mathbf c}\|}<r</math> (see condition above) from the center of the corresponding sphere and its radius is <math>\sqrt{r^2-d^2}</math>. Hence :*<math>{\mathbf x}={\mathbf x}(u,v):= {\mathbf c}(u)-\frac{r(u)\dot{r}(u)}{\|\dot{\mathbf c}(u)\|^2}\dot{\mathbf c}(u) +r(u)\sqrt{1-\frac{\dot{r}(u)^2}{\|\dot{\mathbf c}(u)\|^2}} \big({\mathbf e}_1(u)\cos(v)+ {\mathbf e}_2(u)\sin(v)\big),</math> where the vectors <math>{\mathbf e}_1,{\mathbf e}_2</math> and the tangent vector <math>\dot{\mathbf c}/\|\dot{\mathbf c}\|</math> form an orthonormal basis, is a parametric representation of the canal surface.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf ''Geometry and Algorithms for COMPUTER AIDED DESIGN''], p. 117</ref>
For <math>\dot{r}=0</math> one gets the parametric representation of a '''pipe''' surface: :* <math>{\mathbf x}={\mathbf x}(u,v):= {\mathbf c}(u)+r\big({\mathbf e}_1(u)\cos(v)+ {\mathbf e}_2(u)\sin(v)\big).</math>
300px|thumb|pipe knot 300px|thumb|canal surface: Dupin cyclide
== Examples == :a) The first picture shows a canal surface with :#the helix <math>(\cos(u),\sin(u), 0.25u), u\in[0,4]</math> as directrix and :#the radius function <math>r(u):= 0.2+0.8u/2\pi</math>. :#The choice for <math>{\mathbf e}_1,{\mathbf e}_2</math> is the following: ::<math>{\mathbf e}_1:=(\dot{b},-\dot{a},0)/\|\cdots\|,\ {\mathbf e}_2:= ({\mathbf e}_1\times \dot{\mathbf c})/\|\cdots\|</math>. :b) For the second picture the radius is constant:<math>r(u):= 0.2</math>, i. e. the canal surface is a pipe surface. :c) For the 3. picture the pipe surface b) has parameter <math>u\in[0,7.5]</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).
== See also == * Parallel surface
== References == {{Reflist}}
== Further reading == *{{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 = [https://archive.org/details/geometryimaginat00davi_0/page/219 219] | isbn = 0-8284-1087-9}}
== External links == *[http://www.dmg.tuwien.ac.at/peternell/canalsurf.pdf M. Peternell and H. Pottmann: ''Computing Rational Parametrizations of Canal Surfaces'']
Category:Surfaces