Confocal populations

Abstract

With regard to the theory of confocal sets, the conclusion drawn shows that the set of surfaces derived from the imaginary hyperbola taken as the focal curve does not represent anything new, and was included in those that were derived on the basis of the real hyperbola. If we take into account that in the general case we have two focal curves connected by the main axis on two mutually perpendicular planes of symmetry, one of which is an ellipse and the other a hyperbola, that on the third plane of symmetry the focal curve can be neither an ellipse nor hyperbola, and, as it now turns out, an imaginary hyperbola, then it remains possible to assume only an imaginary ellipse, which is where the derivation of focal curves ends. In conclusion, we note that it is possible to derive involutions on the plane at infinity; since for it three normally conjugate rays are projected from any point, then the corresponding projectivity curve is an imaginary circle, and this is the case for all confocal populations in space.