Английская Википедия: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
- <math>\Phi_c: f({\mathbf x},c)=0 , c\in [c_1,c_2]</math>,
two neighboring surfaces <math>\Phi_c</math> and <math>\Phi_{c+\Delta c}</math> intersect in a curve that fulfills the equations
- <math> f({\mathbf x},c)=0</math> and <math>f({\mathbf x},c+\Delta c)=0</math>.
For the limit <math>\Delta c \to 0</math> one gets <math>f_c({\mathbf x},c)= \lim_{\Delta c \to \ 0} \frac{f({\mathbf x},c)-f({\mathbf x},c+\Delta c)}{\Delta c}=0</math>. The last equation is the reason for the following definition.
- Let <math>\Phi_c: f({\mathbf x},c)=0 , c\in [c_1,c_2]</math> be a 1-parameter pencil of regular implicit <math>C^2</math> surfaces (<math>f</math> being at least twice continuously differentiable). The surface defined by the two equations
- <math> f({\mathbf x},c)=0, \quad f_c({\mathbf x},c)=0 </math>
is the envelope of the given pencil of surfaces.[1]
Canal surface
Let <math>\Gamma: {\mathbf x}={\mathbf c}(u)=(a(u),b(u),c(u))^\top</math> be a regular space curve and <math>r(t)</math> a <math>C^1</math>-function with <math>r>0</math> and <math>|\dot{r}|<\|\dot{\mathbf c}\|</math>. 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
- <math>f({\mathbf x};u):= \big\|{\mathbf x}-{\mathbf c}(u)\big\|^2-r^2(u)=0</math>
is called a canal surface and <math>\Gamma</math> its directrix. If the radii are constant, it is called a pipe surface.
Parametric representation of a canal surface
The envelope condition
- <math>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</math> of the canal surface above is for any value of <math>u</math> the equation of a plane, which is orthogonal to the tangent <math>\dot{\mathbf c}(u)</math> 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 <math>u</math>) has the distance <math>d:=\frac{r\dot{r}}{\|\dot{\mathbf c}\|}<r</math> (see condition above) from the center of the corresponding sphere and its radius is <math>\sqrt{r^2-d^2}</math>. Hence
- <math>{\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),</math> where the vectors <math>{\mathbf e}_1,{\mathbf e}_2</math> and the tangent vector <math>\dot{\mathbf c}/\|\dot{\mathbf c}\|</math> form an orthonormal basis, is a parametric representation of the canal surface.[2]
For <math>\dot{r}=0</math> one gets the parametric representation of a pipe surface:
- <math>{\mathbf x}={\mathbf x}(u,v):=
{\mathbf c}(u)+r\big({\mathbf e}_1(u)\cos(v)+ {\mathbf e}_2(u)\sin(v)\big).</math>
Examples
- a) The first picture shows a canal surface with
- the helix <math>(\cos(u),\sin(u), 0.25u), u\in[0,4]</math> as directrix and
- the radius function <math>r(u):= 0.2+0.8u/2\pi</math>.
- The choice for <math>{\mathbf e}_1,{\mathbf e}_2</math> is the following:
- <math>{\mathbf e}_1:=(\dot{b},-\dot{a},0)/\|\cdots\|,\
{\mathbf e}_2:= ({\mathbf e}_1\times \dot{\mathbf c})/\|\cdots\|</math>.
- b) For the second picture the radius is constant:<math>r(u):= 0.2</math>, i. e. the canal surface is a pipe surface.
- c) For the 3. picture the pipe surface b) has parameter <math>u\in[0,7.5]</math>.
- 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
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