Confocal aggregates

Abstract

With regard to the theory of confocal aggregates, the conclusion drawn shows that the aggregate of surfaces derived from an imaginary hyperbola taken as the focal curve does not represent anything new, and is included among those derived on the basis of a real hyperbola. If we take into account that, in the general case, we have, linked by the principal axis, two focal curves in two mutually perpendicular planes of symmetry, one of which is an ellipse and the other one a hyperbola, and that in the third plane of symmetry the focal curve can be neither an ellipse nor a hyperbola, and, as it now turns out, an imaginary hyperbola, then the only remaining possibility is to admit an imaginary ellipse, thereby completing the derivation of focal curves. In conclusion, we note that involutions can also be derived on the plane at infinity; since from any point three normally conjugate rays are projected onto it, the corresponding projectivity curve is an imaginary circle, and this is the case for any confocal aggregates in space.

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