Conics In Space Research Articles

The conic as a space element

In this article it is proposed to determine a system of coordinates for the conic in space of three dimensions, analogous to the line coordinates of Plucker, and the similar systems for the circle in two and in three dimensions; and to study systems of conics by means of these coordinates. The conic in space has been considerably studied in the last sixty years, but no systematic theory has been developed. In 1908, the Belgian Royal Academy announced the oWer of a prize for a discussion of the subject, but so far as the writer knows, no award was made. It will be of illterest to summarize briefly the work that has been done in this field. In 1861 GrunertS treated the general theory of the conic as a space curve, in connection with the study of planetary orbits. Defining a conic by choosing arbitrarily a line as directrix, a point as focus, and the eccentricity, he deduces a number of analytic relations. He also treats a number of other problems, chiefly of interest to tlle astronomer. About the same time Chaslest laid the foundations of a theory of characteristics of conics in space, analogous to the well-known theory for conics in the plane. Hierholzer§ and Luroth 1I discuss similar problenls. The earliest attempt to determine a set of coordinates for a conic is doubtless that of Spottiswoode.lT His coordinates are not the simplest nor the most natural ones, howeser. NVe shall toucll on them later (§ 3, 6, footllote). A treatment somewhat analogous to parts of ours is given by P. van Geer.** His methods, honTever, are clumsy, and such of his results as are of value +ve can obtain by more direct methods.