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Polarities with respect to a triangle. (Polaritiés définies par un triangle.) (French. English summary) Zbl 1377.51013
Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2010–2011. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 29, 51-71 (2011).
Summary: A polarity of a projective plane is a map, often involutive, sending a generic point to a generic line and vice versa. The most classical polarity is the polarity with respect to a conic section, but others exists: the harmonic polarity with respect to a triangle, the polarities with respect to a higher degree algebraic curve, the polarity with respect to a convex.
In this article we introduce a polarity with respect to a triangle of the projective plane which is motivated by a question on duality of projective frames. We show that the four polarities above that apply to a triangle all coincide in this case. This result is an occasion to revue nice concepts in projective geometry, linear algebra and convex geometry.
For the entire collection see [Zbl 1356.35008].
51N15 Projective analytic geometry
51A05 General theory of linear incidence geometry and projective geometries
53A20 Projective differential geometry
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