# Channel surface

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:

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.

## 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} }}\|}} (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 an orthonormal basis, is a parametric representation of the canal surface.

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

• ${\mathbf {x} }={\mathbf {x} }(u,v):={\mathbf {c} }(u)+r{\big (}{\mathbf {e} }_{1}(u)\cos(v)+{\mathbf {e} }_{2}(u)\sin(v){\big )}.$ ## 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).