Virtual Models of Conic Sections

People know and use conic sections for a long time. They observe these curves in many situations: the parabolic trajectory of a thrown stone, the circular waves when the stone falls in calm water, and the elliptical shadows of round objects during sunsets (see Fig. 1). Conic sections were and still are one of the most favorite objects of mathematical study and education. Students spend hours in the classroom working with circles, ellipses, parabolas, and hyperbolas. They are presented with a concentrated view about these curves, a view that has been distilled for hundreds of years. Although mathematically correct, this view may not lead to complete rationalization, because it might be hard for students to project mathematical ideas into something more comprehensible from their everyday life. A preliminary informal inquiry showed that it is difficult for many students to identify conic sections in a non-classroom environment. This triggered the creation of visualization tools that could introduce these curves from various perspectives. These tools are the focus of this paper. Traditionally, conic sections are described as intersections of a plane and a cone (Downs 1993). Searching the WWW reveals that this is the predominant description of conic sections, independent on whether materials describe mathematical concepts in simple language (like Math2.org’s “A conic section is the intersection of a plane and a cone”) or use scientific terminology (like Wolfram MathWorld’s “The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone”).