Ingeometry andtopology, achannel orcanal surface is a surface formed as theenvelope of a family ofspheres whose centers lie on a spacecurve, itsdirectrix. If the radii of the generating spheres are constant, the canal surface is called apipe 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 anorthographic projection its contour curve can be drawn as the envelope of circles.

• In technical area canal surfaces can be used forblending surfaces smoothly.

Given the pencil ofimplicit surfaces

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

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

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

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

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

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

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

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. 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 acanal surface and${\displaystyle \mathrm{\Gamma }}$ itsdirectrix. If the radii are constant, it is called apipe 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 (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 anorthonormal basis, is a parametric representation of the canal surface.^[2]

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

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

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 aDupin cyclide (canal surface).

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

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

• Surfaces

• Articles with short description
• Short description matches Wikidata