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
pech@pf.jcu.cz
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
weissgunter@gmx.at
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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