Channel surface

Short description: Surface formed from spheres centered along a curve

canal surface: directrix is a helix, with its generating spheres

pipe surface: directrix is a helix, with generating spheres

pipe surface: directrix is a helix

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

${\mathrm{\Phi }}_c:f(\mathbf{𝐱},c)=0,c\in [c_1,c_2]$,

two neighboring surfaces ${\mathrm{\Phi }}_c$ and ${\mathrm{\Phi }}_{c+\mathrm{\Delta }c}$ intersect in a curve that fulfills the equations

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

For the limit $\mathrm{\Delta }c\to 0$ one gets $f_c(\mathbf{𝐱},c)=\underset{\mathrm{\Delta }c\to 0}{\lim }\frac{f(\mathbf{𝐱},c)-f(\mathbf{𝐱},c+\mathrm{\Delta }c)}{\mathrm{\Delta }c}=0$. The last equation is the reason for the following definition.

• Let ${\mathrm{\Phi }}_c:f(\mathbf{𝐱},c)=0,c\in [c_1,c_2]$ 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

  $f(\mathbf{𝐱},c)=0,f_c(\mathbf{𝐱},c)=0$

is the envelope of the given pencil of surfaces.

Canal surface

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

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

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

Parametric representation of a canal surface

The envelope condition

$f_u(\mathbf{𝐱},u)=2(-(\mathbf{𝐱}-\mathbf{𝐜}(u){)}^{\mathrm{\top }}\dot{\mathbf{𝐜}}(u)-r(u)\dot{r}(u))=0$

of the canal surface above is for any value of $u$ the equation of a plane, which is orthogonal to the tangent $\dot{\mathbf{𝐜}}(u)$ 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 $d:=\frac{r\dot{r}}{\Vert \dot{\mathbf{𝐜}}\Vert }<r$ (see condition above) from the center of the corresponding sphere and its radius is $\sqrt{r^2-d^2}$. Hence

• $\mathbf{𝐱}=\mathbf{𝐱}(u,v):=\mathbf{𝐜}(u)-\frac{r(u)\dot{r}(u)}{\Vert \dot{\mathbf{𝐜}}(u){\Vert }^2}\dot{\mathbf{𝐜}}(u)+r(u)\sqrt{1-\frac{\dot{r}(u{)}^2}{\Vert \dot{\mathbf{𝐜}}(u){\Vert }^2}}({\mathbf{𝐞}}_1(u)\cos (v)+{\mathbf{𝐞}}_2(u)\sin (v)),$

where the vectors ${\mathbf{𝐞}}_1,{\mathbf{𝐞}}_2$ and the tangent vector $\dot{\mathbf{𝐜}}/\Vert \dot{\mathbf{𝐜}}\Vert$ form an orthonormal basis, is a parametric representation of the canal surface.

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

• $\mathbf{𝐱}=\mathbf{𝐱}(u,v):=\mathbf{𝐜}(u)+r({\mathbf{𝐞}}_1(u)\cos (v)+{\mathbf{𝐞}}_2(u)\sin (v)).$

pipe knot

canal surface: Dupin cyclide

Examples

a) The first picture shows a canal surface with

1. the helix $(\cos (u),\sin (u),0.25u),u\in [0,4]$ as directrix and
2. the radius function $r(u):=0.2+0.8u/2\pi$.
3. The choice for ${\mathbf{𝐞}}_1,{\mathbf{𝐞}}_2$ is the following:

${\mathbf{𝐞}}_1:=(\dot{b},-\dot{a},0)/\Vert \cdots \Vert ,{\mathbf{𝐞}}_2:=({\mathbf{𝐞}}_1\times \dot{\mathbf{𝐜}})/\Vert \cdots \Vert$.

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 $u\in [0,7.5]$.

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).

References

• Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed.). Chelsea. p. 219. ISBN 0-8284-1087-9. https://archive.org/details/geometryimaginat00davi_0. 

External links

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