Elementary Constructions of conic sections

Abstract

In classical geometry, there is no such well-known and much-studied topic as the construction of conic sections (or briefly conics) from its five points. Its importance in many applications of mechanical engineering, civil engineering and architectural engineering, as well as other applied sciences is clear. The beauty of the topic is that it raises difficult questions that can be approached with basic tools. In this article, we provide constructions (and corresponding theories) that can be taught to high school and university students without knowledge of projective geometry. For this, we recall some important facts about conic sections that can be found in the rich literature. We use the concepts of power of a point on a circle, similarity, orthogonal affinity and inversion. We also mention famous constructions related to our questions. We begin our article at this point, where the standard teaching ends the discussion of conic sections. We therefore assume that the reader knows the basic definitions and constructions of conics, the concepts of focus, axis, tangent, leading circle and leading line.