The Introduction of Metric by the Use of Conics

It has been suggested in several publications [3, 4, 5, 7] that the teaching of systematic deductive geometry should begin with plane affine geometry. Reports of experiments in this direction are already available. I will not discuss the advantages or the disadvantages. Instead, I shall consider a consequence of this approach to geometry. How can the metric concepts (distance, orthogonality) be introduced in a well-motivated way if the plane affine geometry is already developed to such an extent that it can be described by a two dimensional vector space over the field of the reals. Of course the students already have intuitive and experimental knowledge of the simplest properties of the circle. So it might be appropriate to investigate certain curves in the affine plane with some of these properties and then to choose one of these curves as unit circle, defining distance and orthogonality by it. The lines which are described by linear forms obviously don’t have the necessary properties of such curves so one is motivated to try the quadratic forms derived from the symmetric bilinear forms. In this way one gets to the curves of second degree with a center. In the title of this paper we refer to them as “conics”. Thus creation of the metric notions as supplementary concepts not readily available from the outset opens to the student the possibility of metrics other than the usual Euclidean one. In particular, an indefinite quadratic form gives the Minkowski metric which plays an important role in the theory of relativity. Using conics in connection with the bilinear forms and related to the introduction of metric, in my opinion gives these objects the right place in contemporary mathematics and saves them from being junked, a fate they might receive in retaliation for the overweight they have had in traditional teaching.