Channel surface

Given the pencil of implicit surfaces

:$\backslash Phi\_c:\; f(\{\backslash mathbf\; x\},c)=0\; ,\; c\backslash in\; [c\_1,c\_2]$,

two neighboring surfaces $\backslash Phi\_c$ and

$\backslash Phi\_\{c+\backslash Delta\; c\}$ intersect in a curve that fulfills the equations

:$f(\{\backslash mathbf\; x\},c)=0$ and $f(\{\backslash mathbf\; x\},c+\backslash Delta\; c)=0$.

For the limit $\backslash Delta\; c\; \backslash to\; 0$ one gets

$f\_c(\{\backslash mathbf\; x\},c)=\; \backslash lim\_\{\backslash Delta\; c\; \backslash to\; \backslash \; 0\}\; \backslash frac\{f(\{\backslash mathbf\; x\},c)-f(\{\backslash mathbf\; x\},c+\backslash Delta\; c)\}\{\backslash Delta\; c\}=0$.

The last equation is the reason for the following definition.

• Let $\backslash Phi\_c:\; f(\{\backslash mathbf\; x\},c)=0\; ,\; c\backslash in\; [c\_1,c\_2]$ 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
• :$f(\{\backslash mathbf\; x\},c)=0,\; \backslash quad\; f\_c(\{\backslash mathbf\; x\},c)=0$

is the envelope of the given pencil of surfaces.[http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf Geometry and Algorithms for COMPUTER AIDED DESIGN], p. 115

Let $\backslash Gamma:\; \{\backslash mathbf\; x\}=\{\backslash mathbf\; c\}(u)=(a(u),b(u),c(u))^\backslash top$ be a regular space curve and $r(t)$ a $C^1$-function with $r>0$ 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

:$f(\{\backslash mathbf\; x\};u):=\; \backslash big\backslash |\{\backslash mathbf\; x\}-\{\backslash mathbf\; c\}(u)\backslash big\backslash |^2-r^2(u)=0$

is called a canal surface and $\backslash Gamma$ its directrix. If the radii are constant, it is called a pipe surface.

The envelope condition

:$f\_u(\{\backslash mathbf\; x\},u)=$

2\Big(-\big({\mathbf x}-{\mathbf c}(u)\big)^\top\dot{\mathbf c}(u)-r(u)\dot{r}(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

$\backslash dot\{\backslash mathbf\; c\}(u)$ 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

from the center of the corresponding sphere and its radius is $\backslash sqrt\{r^2-d^2\}$. Hence

:*$\{\backslash mathbf\; x\}=\{\backslash mathbf\; x\}(u,v):=$

{\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),

where the vectors $\{\backslash mathbf\; e\}\_1,\{\backslash mathbf\; e\}\_2$ and the tangent vector $\backslash dot\{\backslash mathbf\; c\}/\backslash |\backslash dot\{\backslash mathbf\; c\}\backslash |$ form an orthonormal basis, is a parametric representation of the canal surface.[http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf Geometry and Algorithms for COMPUTER AIDED DESIGN], p. 117

For $\backslash dot\{r\}=0$ one gets the parametric representation of a pipe surface:

:* $\{\backslash mathbf\; x\}=\{\backslash mathbf\; x\}(u,v):=$

{\mathbf c}(u)+r\big({\mathbf e}_1(u)\cos(v)+ {\mathbf e}_2(u)\sin(v)\big).

File:Pipe-knot-s.svg

File:Cyclid-s.svg

:a) The first picture shows a canal surface with

:#the helix $(\backslash cos(u),\backslash sin(u),\; 0.25u),\; u\backslash in[0,4]$ as directrix and

:#the radius function $r(u):=\; 0.2+0.8u/2\backslash pi$.

:#The choice for $\{\backslash mathbf\; e\}\_1,\{\backslash mathbf\; e\}\_2$ is the following:

::$\{\backslash mathbf\; e\}\_1:=(\backslash dot\{b\},-\backslash dot\{a\},0)/\backslash |\backslash cdots\backslash |,\backslash$

{\mathbf e}_2:= ({\mathbf e}_1\times \dot{\mathbf c})/\|\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 $u\backslash in[0,7.5]$.

: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).

{{Reflist}}

• {{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}}

• [http://www.dmg.tuwien.ac.at/peternell/canalsurf.pdf M. Peternell and H. Pottmann: Computing Rational Parametrizations of Canal Surfaces]

Category:Surfaces