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.
 * 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. is the envelope of the given pencil of surfaces.
 * 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
 * $$ f({\mathbf x},c)=0, \quad   f_c({\mathbf x},c)=0 $$

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).