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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: * 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 :$\Phi_c: f\left(,c\right)=0 , c\in_1,c_2/math>, two neighboring surfaces\Phi_cand\Phi_intersect in a curve that fulfills the equations :f\left(,c\right)=0andf\left(,c+\Delta c\right)=0. For the limit\Delta c \to 0one getsf_c\left(,c\right)= \lim_ \frac=0. The last equation is the reason for the following definition. * Let\Phi_c: f\left(,c\right)=0 , c\in_1,c_2/math> be a 1-parameter pencil of regular implicitC^2surfaces \left(fbeing at least twice continuously differentiable\right). The surface defined by the two equations *:f\left(,c\right)=0, \quad f_c\left(,c\right)=0is the envelope of the given pencil of surfaces.$

Canal surface

Let $\Gamma: =\left(u\right)=\left(a\left(u\right),b\left(u\right),c\left(u\right)\right)^\top$ be a regular space curve and $r\left(t\right)$ a $C^1$-function with $r>0$ and $|\dot|<\|\dot\|$. 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 :$f\left(;u\right):= \big\|-\left(u\right)\big\|^2-r^2\left(u\right)=0$ 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 :$f_u\left(,u\right)= 2\Big\left(-\big\left(-\left(u\right)\big\right)^\top\dot\left(u\right)-r\left(u\right)\dot\left(u\right)\Big\right)=0$ of the canal surface above is for any value of $u$ the equation of a plane, which is orthogonal to the tangent $\dot\left(u\right)$ 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
p. 117 For $\dot=0$ one gets the parametric representation of a pipe surface: :* $=\left(u,v\right):= \left(u\right)+r\big\left(_1\left(u\right)\cos\left(v\right)+ _2\left(u\right)\sin\left(v\right)\big\right).$

Examples

:a) The first picture shows a canal surface with :#the helix $\left(\cos\left(u\right),\sin\left(u\right), 0.25u\right), u\in,4/math> as directrix and :#the radius functionr\left(u\right):= 0.2+0.8u/2\pi. :#The choice for_1,_2is the following: ::_1:=\left(\dot,-\dot,0\right)/\|\cdots\|,\ _2:= \left(_1\times \dot\right)/\|\cdots\|. :b\right) For the second picture the radius is constant:r\left(u\right):= 0.2, i. e. the canal surface is a pipe surface. :c\right) For the 3. picture the pipe surface b\right) has parameteru\in,7.5/math>. :d\right) The 4. picture shows a pipe knot. Its directrix is a curve on a torus :e\right) The 5. picture shows aDupin cyclide\left(canal surface\right).$

References

*{{cite book |author1= Hilbert, David |author-link= David Hilbert |author2=Cohn-Vossen, Stephan | title = Geometry and the Imagination |url= https://archive.org/details/geometryimaginat00davi_0|url-access= registration| edition = 2nd | year = 1952 | publisher = Chelsea | page
219
| isbn = 0-8284-1087-9