Abstract
The exponential pencil G_lambda :=G_1(G_0^{-1}G_1)^{lambda -1}, generated by two conics G_0,G_1, carries a rich geometric structure: It is closed under conjugation, it is compatible with duality and projective mappings, it is convergent for lambda rightarrow pm infty or periodic, and it is connected in various ways with the linear pencil g_lambda =lambda G_1+(1-lambda )G_0. The structure of the exponential pencil can be used to characterize the position of G_0 and G_1 relative to each other.
Highlights
The linear pencil gλ = λG1 + (1 − λ)G0, λ ∈ R, of two circles or conics G0 and G1 is an extremely useful tool in the study of the geometry of circles and of conic sections, or, in higher dimensions, of quadrics
The exponential pencil Gλ := G1(G−0 1G1)λ−1, generated by two conics G0, G1, carries a rich geometric structure: It is closed under conjugation, it is compatible with duality and projective mappings, it is convergent for λ → ±∞ or periodic, and it is connected in various ways with the linear pencil gλ = λG1 + (1 − λ)G0
The structure of the exponential pencil can be used to characterize the position of G0 and G1 relative to each other
Summary
The linear pencil gλ = λG1 + (1 − λ)G0, λ ∈ R, of two circles or conics G0 and G1 is an extremely useful tool in the study of the geometry of circles and of conic sections, or, in higher dimensions, of quadrics. 2.1 and 3), and the linear pencil does, in general, not exist as real conics for all λ ∈ R. It turns out, that this pencil has a remarkable spectrum of geometric properties, which we study in Sect. 4 we classify the exponential pencils according to the relative position of the generating conics. We start with some preliminary remarks to set the stage and to fix the notation
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