Channel surface

canalsurface: directrix is a helix, with its generating spheres

pipe surface: directrix is a helix, with generating spheres

pipe surface: directrix is a helix

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 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

$\Phi_c: f({\mathbf x},c)=0 , c\in [c_1,c_2]$.

Two neighboring surfaces $\Phi_c$ and $\Phi_{c+\Delta c}$ intersect in a curve that fulfills the equations

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

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

• Let be $\Phi_c: f({\mathbf x},c)=0 , c\in [c_1,c_2]$ a 1-parameter pencil of regular implicit $C^2$ - surfaces ($f$ is at least twice continuously differentiable). The surface defined by the two equations

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

is the envelope of the given pencil of surfaces.^[1]

Canal surface

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

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

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

Parametric representation of a canal surface

The envelope condition

$f_u({\mathbf x},u):= 2\Big(\big({\mathbf x}-{\mathbf c}(u)\big)\dot{\mathbf c}(u)-r(u)\dot{r}(u)\Big)=0$,

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

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

where the vectors ${\mathbf e}_1,{\mathbf e}_2$ and the tangenten vector $\dot{\mathbf c}$ form a orthonormal basis, is a parametric representation of the canal surface.^[2]

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

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

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:

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

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 $u\in[0,7.5]$.

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

1. Geometry and Algorithms for COMPUTER AIDED DESIGN, p. 115
2. Geometry and Algorithms for COMPUTER AIDED DESIGN, p. 117

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

External links

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