Circle and Hyperbola As Lighthouse Curves

Circle and Hyperbola As Lighthouse Curves
A Mathematical Droodle

Two lines rotate with constant angular velocities around two points in a plane. (An apt metaphore of two lighthouses was invented by R. K. Guy of the University of Calgary.) The point illuminated by two beams traces a curve. What is it?

Equal angular velocities:

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Opposite angular velocities

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Explanation

For circles, the explanation is based on the equality of the inscribed angles subtended by the same arc.

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Circle and Hyperbola As Lighthouse Curves A Mathematical Droodle

Equal angular velocities:

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

Opposite angular velocities

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

Circle and Hyperbola As Lighthouse Curves
A Mathematical Droodle