Pencils of Frégier Conics

Author:

Odehnal Boris1ORCID

Affiliation:

1. University of Applied Arts Vienna

Abstract

For each point P on a conic c, the involution of right angles at P induces an elliptic involution on c whose center F is called the Frégier point of P. Replacing the right angles at P between assigned pairs of lines with an arbitrary angle phi yields a projective mapping of lines in the pencil about P, and thus, on c. The lines joining corresponding points on c do no longer pass through a single point and envelop a conic f which can be seen as the generalization of the Frégier point and shall be called a generalized Frégier conic. By varying the angle, we obtain a pencil of generalized Frégier conics which is a pencil of the third kind. We shall study the thus defined conics and discover, among other objects, general Poncelet triangle families.

Publisher

Croatian Society for Geometry and Graphics

Subject

General Earth and Planetary Sciences,General Environmental Science

Reference17 articles.

1. Halbeisen, L., Hungerbühler, N., The exponential pencil of conics, Beitr. Algebra Geom. 59 (2018), 549-571, https://doi.org/10.1007/s13366-017-0375-1

2. Weiss, G., Frégier points revisited, Proceedings of the Czeck-Slovak Conference on Geometry and Graphics 2018, 277-286.

3. Tummers, J.H., Quelques théorèmes par rapport au point de Frégier,Chr. Huygens 9 (1931), 201-205.

4. Schröcker, H.-P., Singular Fégier Conics in Non-Euclidean Geometry, J. Geom. Graph. 21(2) (2017), 201-208.

5. Schröcker, H.-P., A Family of Conics an Three Special Ruled Surfaces, Beitr. Algebra Geom. 42(2) (2001), 531-545.

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