Channel surface

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.

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 r0 and |\dot{r}|. 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}|} (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).

References

References

1. [http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf ''Geometry and Algorithms for COMPUTER AIDED DESIGN''], p. 115
2. [http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf ''Geometry and Algorithms for COMPUTER AIDED DESIGN''], p. 117