Circle and Hyperbola As Lighthouse Curves II

Elsewhere we asked this question: two lines rotate with constant angular velocities around two points in a plane. The point illuminated by two beams traces a curve. What is it?

The applet below presents a stationary configuration. There are two pencils of straight lines that represent successive positions of the rotating lines at equal intervals, the intersections of the matching lines being highlighted.

The explanation is rather simple in case of a circle: the triangle formed by the vertices of the two pencils and the(moving) point of intersection of the corresponding lines has base angles whose sum is fixed making the third angle, at the point of intersection also constant. This is exactly the characterization of inscribed angles subtended the same arc in circles.

Copyright © 1996-2009 Alexander Bogomolny