The exponential pencil of conics
In the paper under review, the authors consider the so-called exponential pencils of conics in the real projective plane. In order to emphasize the authors' approach, we define a conic as an equivalence class of a regular, linear, self-adjoint map \(A :\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}\) with mixed signature (\(A\) has eigenvalues of both signs). One can identify a conic by the set of points \([x]\) such that \(\langle x, Ax \rangle = 0\). Definition 1. Let \(G_{0}, G_{1}\) be two conics. Then \[ G_{\lambda} :=G_{1}(G_{0}^{-1}G_{1})^{\lambda - 1} \text{ with } \lambda \in \mathbb{R} \] is called an exponential pencil generated by \(G_{0}\) and \(G_{1}\) provided that all \(G_{\lambda}\) are symmetric and real. After presenting some simple properties of these exponential pencils of conics, the authors provide a detailed classification of exponential pencils depending on a position of two generating conics (we have exactly 8 possibilities according to \textit{S. Petitjean}'s approach [in: Nonlinear computational geometry. Papers presented at the workshop, Minneapolis, MN, USA, May 29--June 2, 2007. Dordrecht: Springer. 189--220 (2010; Zbl 1190.13004)]). We recall only one result in the case when two conics intersect along four points. Theorem. Let \(G_{0}, G_{1}\) be two conics having four intersection points. Then they generate an exponential conic \(G_{\lambda} = G_{1}(G_{0}^{-1}G_{1})^{\lambda -1}\) iff the common interior of \(G_{1}, G_{0}\) is connected. In this case, the exponential pencil is unique, \(G_{\lambda}\) converges for \(\lambda \rightarrow \pm \infty\) to a line \(\ell^{\pm}\). The family \(G_{\lambda}\) has an envelope \(E\) with asymptotes \(\ell^{\pm}\). Through every exterior point of \(E\) (i.e., points with four tangents), except of the points on \(\ell^{\pm}\), there pass exactly two members of the exponential pencil \(G_{\lambda}\). Each \(G_{\lambda}\) touches a member of the linear pencil \(g_{\lambda} = \lambda G_{1} + (1 - \lambda)G_{0}\) in two first order contact points.
scientific article; zbMATH DE number 6933464