IV order ruled surface with high symmetry and a curve with four cusp points

Abstract

As is known, these fourth-order surfaces can be reproduced by two projective quadratic prima planes. The theory of these surfaces is presented in detailed manuals. In the general case, there is a hexaprima of points on the surface at which two rays of the surface intersect. Planes passing through such a pair of rays can be considered tangent planes, and since a flat section of a surface in the general case is a curve of order IV, it is clear that if a plane passes through one of the rays of the surface, then this curve splits into this ray and a curve of III order, and if the plane passes through a pair of rays, then the same curve breaks up into this pair and another conoprima; from here we see that the tangent planes passing through pairs of rays of the surface intersect this surface while still in conoprima.