Bisoptic curves of hyperbolas

Abstract

Given a hyperbola, we study its bisoptic curves, i.e. the geometric locus of points through which passes a pair of tangents making a fixed angle θ or 180° − θ. This question has been addressed in a previous paper for parabolas and for ellipses, showing hyperbolas and spiric curves, respectively. Here the requested geometric locus can be empty. If not, it is a punctured spiric curve, and two cases occur: the curve can have either one loop or two loops. Finally, we reconstruct explicitly the spiric curve as the intersection of a plane with a self-intersecting torus.

Keywords:

• bisoptic curves
• conic sections
• toric sections

Notes

1. Freely downloadable from www.geogebra.org.

2. The authors recommend to experiment the situation using a software for dynamical geometry. They did it using the GeoGebra (www.geogebra.org) program.

3. The file of a GeoGebra animation can be received on request from one of the authors.

4. The animation has been prepared for Derive 6 and uses the slider bar. The file OpticCurve-hyperbola.dfw can be requested from one of the authors.