Channel surface

 ( Surfaces ) 

In geometry and topology, a channel or canal surface is a surface formed as the envelope of a family of whose centers lie on a space curve, its Generatrix. 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.

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Envelope of a pencil of implicit surfaces Given the pencil of

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

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. Geometry and Algorithms for COMPUTER AIDED DESIGN, p. 115

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Canal surface 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.

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Parametric representation of a canal 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

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

{\mathbf c}(u)-\frac{r(u)\dot{r}(u)}{\|\dot{\mathbf c}(u)\|^2}\dot{\mathbf c}(u) \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. 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).

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Examples

a) The first picture shows a canal surface with

#the helix $(\backslash cos(u),\backslash sin(u),\; 0.25u),\; u\backslash in0,4$ 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 $u\backslash in0,7.5$.

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).

• (2025). 9780828410878, Chelsea. . ISBN 9780828410878

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External links

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

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