Jakob Steiner's Construction of Conics Revisited

P. Pech ; G. Weiss

Journal for geometry and graphics, Tome 23 (2019) no. 2, pp. 189-199

Résumé

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.

Détail

Classification : 51M15, 51N15
Mots-clés : Conic, real projective plane, projectivity, Steiner's generation of a conic