Die zweifachen Blutelschen Kegelschnittflächen

Abstract

A surface Φ in projective space generated by a one parameter family of conics is called a conic surface of Blutel if the tangent planes of Φ taken along a generating conic, envelop a quadratic cone. If the conjugate curves (with respect to the generating conics) are conics, too, we call Φ a two-fold Blutel's conic surface. In an earlier paper [4] it was shown that the planes of both conic families, the generating and the conjugate one, belong to a pencil, each. The present paper completes these investigations by integrating the derivative equations (3), (8), (9), (10). As a final result, a complete classification of all these surfaces is given. They are all algebraic of at most fourth order and furthermore—besides the quadrics and certain degenerate cases—they are complex projectively equivalent to the cyclides of Dupin.