Channel surface

Given the pencil of implicit surfaces

${\displaystyle \Phi _{c}:f({\mathbf {x} },c)=0,c\in [c_{1},c_{2}]}$,

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

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

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

• Let ${\displaystyle \Phi _{c}:f({\mathbf {x} },c)=0,c\in [c_{1},c_{2}]}$ 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

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

is the envelope of the given pencil of surfaces.[1]

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

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

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

The envelope condition

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

of the canal surface above is for any value of ${\displaystyle u}$ 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 ${\displaystyle d:={\frac {r{\dot {r}}}{\|{\dot {\mathbf {c} }}\|}}<r}$ (see condition above) from the center of the corresponding sphere and its radius is ${\displaystyle {\sqrt {r^{2}-d^{2}}}}$. Hence

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

where the vectors ${\displaystyle {\mathbf {e} }_{1},{\mathbf {e} }_{2}}$ and the tangent vector ${\displaystyle {\dot {\mathbf {c} }}/\|{\dot {\mathbf {c} }}\|}$ form an orthonormal basis, is a parametric representation of the canal surface.[2]

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

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

pipe knot

canal surface: Dupin cyclide

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:

${\displaystyle {\mathbf {e} }_{1}:=({\dot {b}},-{\dot {a}},0)/\|\cdots \|,\ {\mathbf {e} }_{2}:=({\mathbf {e} }_{1}\times {\dot {\mathbf {c} }})/\|\cdots \|}$.

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).

• 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