Graphical Multiplication Using Straightedge When a Conic Is Given (A Generalization of Möbius’s Observation)

Around 1840, A.F. Möbius discovered a remarkable property of the parabola: the curve drawn in the Cartesian coordinate system can be used to graphically multiply any integers. In this article, we generalize this algorithm in two regards: (1) we consider any conic in the plane without a Cartesian coordinate system; (2) we use an unmarked straightedge (i.e., only lines and intersections may be drawn). At the end of the article, we will discuss a special case in which the general conic section is replaced by a circle. We will determine some metric properties of this model and pose an open question as to whether this model represents a certain form of non-Euclidean geometry.