Channel surface

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

${\Phi }_c:f(𝐱,c)=0,c\in [c_1,c_2]$,

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

$f(𝐱,c)=0$ and $f(𝐱,c+\Delta c)=0$.

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

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

  $f(𝐱,c)=0,f_c(𝐱,c)=0$

is the envelope of the given pencil of surfaces.

Canal surface

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

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

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

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

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

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

where the vectors ${𝐞}_1,{𝐞}_2$ and the tangent vector $\dot{𝐜}/\Vert \dot{𝐜}\Vert$ 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:

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

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 ${𝐞}_1,{𝐞}_2$ is the following:

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

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

• Template:Cite book

External links

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