# Channel surface

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.

## 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 $f_{c}({\mathbf {x} },c)=\lim _{\Delta c\to \ 0}{\frac {f({\mathbf {x} },c)-f({\mathbf {x} },c+\Delta c)}{\Delta c}}=0$. The last equation is the reason for the following definition.
* Let $\Phi _{c}:f({\mathbf {x} },c)=0,c\in [c_{1},c_{2}]$ 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
    $f({\mathbf {x} },c)=0,\quad f_{c}({\mathbf {x} },c)=0$
is the envelope of the given pencil of surfaces.

## Canal surface

Let $\Gamma :{\mathbf {x} }={\mathbf {c} }(u)=(a(u),b(u),c(u))^{\top }$ be 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
    $f({\mathbf {x} };u):={\big \|}{\mathbf {x} }-{\mathbf {c} }(u){\big \|}^{2}-r^{2}(u)=0$
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
    $f_{u}({\mathbf {x} },u)=2{\Big-{\big{\mathbf {x} }-{\mathbf {c} }(u){\big )}^{\top }{\dot {\mathbf {c} }}(u)-r(u){\dot {r}}(u){\Big )}=0$
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 $d:={\frac {r{\dot {r}}}{\|{\dot {\mathbf {c} }}\|}}<r$ (see condition above) from the center of the corresponding sphere and its radius is ${\sqrt {r^{2}-d^{2}}}$. Hence
   * ${\mathbf {x} }={\mathbf {x} }(u,v):={\mathbf {c} }(u)-{\frac {r(u){\dot {r}}(u)}{\|{\dot {\mathbf {c} }}(u)\|^{2}}}{\dot {\mathbf {c} }}(u)+r(u){\sqrt {1-{\frac {{\dot {r}}(u)^{2}}{\|{\dot {\mathbf {c} }}(u)\|^{2}}}}}{\big{\mathbf {e} }_{1}(u)\cos(v)+{\mathbf {e} }_{2}(u)\sin(v){\big )},$
where the vectors ${\mathbf {e} }_{1},{\mathbf {e} }_{2}$ and the tangent vector ${\dot {\mathbf {c} }}/\|{\dot {\mathbf {c} }}\|$ 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
   * the helix $(\cos(u),\sin(u),0.25u),u\in [0,4]$ as directrix and
   * the radius function $r(u):=0.2+0.8u/2\pi$.
   * 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).