Projection pencil of quadrics and Ivory theorem

TL;DR

This paper explores the properties of confocal quadrics in pseudo-Euclidean spaces, clarifying differences from Euclidean confocality and analyzing both regular and singular projection cases.

Contribution

It provides a new formulation of confocality in pseudo-Euclidean spaces and investigates singular cases, extending the understanding beyond classical Euclidean definitions.

Findings

Generalized confocality differs from Euclidean case in pseudo-Euclidean spaces

Regular projection yields classical confocality properties

Singular projections reveal important geometric behaviors

Abstract

We rewrite the property of confocality with respect to a pseudo-Euclidean space. Our observation is that the generalized definition of confocality in $\cite{stachel}$ does not give back to the original definition of confocality of Euclidean conics. The "confocality property" of $\cite{stachel}$ can be get from our definition in the case when the projection transformation is regular, it is the identity transformation of the space. Our examination concentrate to the singular cases, too because they are playing important role in the investigations of essential examples.