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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(,c)=0 , c\in _1,c_2/math>, two neighboring surfaces \Phi_c and \Phi_ intersect in a curve that fulfills the equations : f(,c)=0 and f(,c+\Delta c)=0. For the limit \Delta c \to 0 one gets f_c(,c)= \lim_ \frac=0. The last equation is the reason for the following definition. * Let \Phi_c: f(,c)=0 , c\in _1,c_2/math> 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(,c)=0, \quad f_c(,c)=0 is the envelope of the given pencil of surfaces.


Canal surface


Let \Gamma: =(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|<\|\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(;u):= \big\|-(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(,u)= 2\Big(-\big(-(u)\big)^\top\dot(u)-r(u)\dot(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(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 (see condition above) from the center of the corresponding sphere and its radius is \sqrt. Hence :*=(u,v):= (u)-\frac\dot(u) +r(u)\sqrt \big(_1(u)\cos(v)+ _2(u)\sin(v)\big), where the vectors _1,_2 and the tangent vector \dot/\|\dot\| form an orthonormal basis, is a parametric representation of the canal surface.''Geometry and Algorithms for COMPUTER AIDED DESIGN''
p. 117 For \dot=0 one gets the parametric representation of a pipe surface: :* =(u,v):= (u)+r\big(_1(u)\cos(v)+ _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,4/math> as directrix and :#the radius function r(u):= 0.2+0.8u/2\pi. :#The choice for _1,_2 is the following: ::_1:=(\dot,-\dot,0)/\|\cdots\|,\ _2:= (_1\times \dot)/\|\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,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


*{{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


External links



M. Peternell and H. Pottmann: ''Computing Rational Parametrizations of Canal Surfaces''
Category:Surfaces