
Journal for Geometry and Graphics 23 (2019), No. 2, 189199 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. 810/104, 1040 Vienna, Austria weissgunter@gmx.at We aim at presenting material on conics, which can be used to formulate, e.g., GeoGebra problems for highschool 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 crossratio 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 wellknown 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 socalled secondorder biarcs. Keywords: Conic, real projective plane, projectivity, Steiner's generation of a conic. MSC: 51M15; 51N15 [ Fulltextpdf (2019 KB)] 