In geometry and topology, a channel surface 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:

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]

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
  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).

See also

References

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

Further reading