Channel surface

canalsurface: directrix is a helix, with its generating spheres
pipe surface: directrix is a helix, with generating spheres
pipe surface: directrix is a helix

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.

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

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

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}\|}<r (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 a orthonormal basis, is a parametric representation of the canal surface.[2]

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).
pipe knot
canal surface: Dupin cyclide

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).

References

  1. Geometry and Algorithms for COMPUTER AIDED DESIGN, p. 115
  2. Geometry and Algorithms for COMPUTER AIDED DESIGN, p. 117

External links

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