Channel surface

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:

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.

Given the pencil of implicit surfaces

Two neighboring surfaces ${\displaystyle \Phi _{c}}$ and ${\displaystyle \Phi _{c+\Delta c}}$ intersect in a curve that fulfills the equations

For the limit ${\displaystyle \Delta c\to 0}$ one gets ${\displaystyle f_{c}({\mathbf {x} },c)=\lim _{\Delta \to \ 0}{\frac {f({\mathbf {x} },c)-f({\mathbf {x} },c+\Delta c)}{\Delta c}}=0}$. The last equation is the reason for the following definition

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