Channel surface

Canal surface

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

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

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

Parametric representation of a canal surface

The envelope condition

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

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

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

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

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

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

Examples

a) The first picture shows a canal surface with

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

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

b) For the second picture the radius is constant:${\displaystyle r(u):=0.2}$, i. e. the canal surface is a pipe surface.

c) For the 3. picture the pipe surface b) has parameter ${\displaystyle u\in [0,7.5]}$.

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).