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Journal for Geometry and Graphics 23 (2019), No. 2, 189--199
Copyright Heldermann Verlag 2019

Jakob Steiner's Construction of Conics Revisited

Pavel Pech
Faculty of Education, University of South Bohemia, Jeronýmova 10, 371 15 Ceské Budejovice, Czech Republic

Gunter Weiss
Institute for Geometry, Technical University, 01062 Dresden, Germany
and: Inst. of Discrete Mathematics, University of Technology, Wiedner Hauptstr. 8-10/104, 1040 Vienna, Austria

We aim at presenting material on conics, which can be used to formulate, e.g., GeoGebra problems for high-school and freshmen maths courses at universities. In a (real) projective plane, two pencils of lines, which are projectively related, generate, in general, a conic. This fact due to Jakob Steiner allows to construct points of a conic given by, e.g., 5 points. Hereby the problem of transfering a given cross-ratio of four lines of the first pencil to the corresponding ones in the second pencil occurs. To solve this problem in a graphically simple and uniform way, we propose a method, which uses the well-known fact that a projective mapping from one line or pencil to another always can be decomposed into a product of perspectivities. By extending the presented graphical methods, we also construct tangents and osculating circles at points of a conic. The calculation following the graphic treatment delivers a parametrisation of conic arcs applicable also for so-called second-order biarcs.

Keywords: Conic, real projective plane, projectivity, Steiner's generation of a conic.

MSC: 51M15; 51N15

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