A canal surface containing four straight lines

Abstract

A canal surface is the envelope of spheres with centers traversing a spatial curve called spine curve. The spheres contact the envelope along so-called characteristics, which are circles in general. If a canal surface contains two lines, then the spine curve is located on the bisector of these lines which in the case of skew lines is an orthogonal hyperbolic paraboloid. There are trivial cases of canal surfaces with infinitely many lines, the right cylinders, the right cones, and the one-sheeted hyperboloids of revolution. The only nontrivial case of a canal surface through four straight lines, that are not the limits of characteristics, is related to a Plücker conoid. The four given lines must be concyclic generators, i.e., they intersect each tangent plane of the conoid in four points lying on a circle. We are going to analyse and visualize this par ticular canal surface.