Canal Surfaces and Inversive Geometry Univ. Introduction Canal surfaces are an important tool in geometric modeling, specially in blending and joining surface primitives along circular profiles. These include planes, cones, Dupin cyclides and arbitrary surfaces of revolution.
Planes are spheres with infinit radius and lines are circles with infinite radius.
Thus, talking about spheres and circles means talking about spheres, planes, circles and lines. A canal surface is the envelope of a 1-parametric family of spheres. The envelope is defined as the union of all circles of intersection of infinitessimally neighbouring pairs of spheres. The generalized sterographic projection maps S onto the pole of the hyperplane H with respect to the Moebius Hypersphere. These are mapped onto to linear families of spheres or planes which are classically known as coaxial pencils.