Channel surface

canal surface: directrix is a helix, with its generating spheres

pipe surface: directrix is a helix, with generating spheres

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.

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 $f_{c}({\mathbf {x} },c)=\lim _{\Delta c\to \ 0}{\frac {f({\mathbf {x} },c)-f({\mathbf {x} },c+\Delta c)}{\Delta c}}=0$ . 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.^[1]

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.^[2]