Analytic systems of central conics in space

Abstract

The amount of literature dealing with conic sections, individual curves and systems of curves, in one plane, is vast. When however we are dealing with a number of conics, not in the same plane, the situation is quite different. Certain figures, as the focal conics of a set of confocal quadrics, are familiar enough, but very little has been done in the way of a systematic study of more general systems. There are some studies carried out with the aid of purely synthetic methods; the algebraic or analytic treatment lags behind. The first writer to suggest a reasonable set of coordinates for a conic in space was Spottiswoode(l). The totality of straight lines that intersect a conic in three-space generates a very special sort of quadratic complex. The coefficients determining the equation of this complex, when a straight line has the usual Plucker line coordinates, may be taken as the coordinates of the conic, a clumsy enough system. A much better technique, perhaps the best for algebraic purposes, was developed by Johnson(2). Here a conic is looked upon not as a locus, but as the envelop of its tangent planes. Thus its tangential equation aiiuiuuO =0 gives ten homogeneous coordinates, connected by a quartic identity aii = aii, aiil = 0 i, j = 1, 2, 3, 4.