Note on a certain type of parabola

Abstract

In the development of the geometry of the complex domain, there has been a tendency to pass over problems of a relatively elementary character, despite their central position and importance. It is the purpose of this note to call attention to one of these problems, namely that of the classification of complex conies with respect to the group of complex rigid motions. A systematic study of this problem brings to light two special types of non-degenerate conies: the non-degenerate central conies which contain just one circular point at infinity, and the non-degenerate parabolas which are tangent to the line at infinity at a circular point. These special conies have already been considered, from a different point of view, in this Bulletin, f It is not our intention to go into detail, either in connection with the general problem of classification or in consideration of the special conies. We propose merely to discuss certain particularly striking properties of the special parabolas. I t is evident that the complex conic (1) Ax + Bxy + Cy + Dx + Ey + F = 0,