Canal surfaces 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. These are circles in general. When in limiting poses the spheres become planes, then the characteristics are lines. We focus on cases where the envelope contains lines that contact all spheres and, consequently, are no characteristics. Trivial cases of canal surfaces with infinitely many lines are the right cylinders and cones and the one-sheeted hyperboloids of revolution. If the number of non-characteristic lines on the canal surface is finite, then it is less or equal four. The maximum holds if the four lines are located on a Plücker conoid and intersect each tangent plane of the conoid in concyclic points. We are going to analyse these particular canal surfaces which in general are hard to visualize due to their singularities: They contain cuspidal edges, and the points of the lines are biplanar or u niplanar points of the algebraically closed surface. Symmetric versions are easier to grasp. Even parabolic Dupin ring cyclides and needle cyclides are included as limiting cases when given lines coincide.