From an Ellipse To a Hyperbola (or, There and Back Again)

As my Precalculus students begin to explore conic sections, a related activity that is a lot of fun is to use the concept of locus to generate them. A locus is the set of all points satisfying some condition. For example, the locus of all points equidistant from a fixed point would be a circle. The locus of all points equidistant from a line and a point not on that line would be a parabola (but you already knew that if you had seen this earlier post!).

I used to use waxpaper folding to create repeated perpendicular bisectors that eventually resulted in conic sections. It was very tedious, but now Geometer’s Sketchpad can do it in seconds, as well as animate the “creases”.

Here’s the basic setup, if you want to do it for yourself:

The blue line is the perpendicular bisector of segment DE. Point E moves around the circle, while point D moves bidirectionally along segment BC. I had Sketchpad trace the blue line (as well as change its color based upon the length of segment DE). You can watch the results in the video below. By the way, Screencast-O-Matic has a new feature allowing the screencaster to add sound directly from your computer. I was in an ’80s mood, so I set this video to Jan Hammer’s song, Evan, from the Miami Vice soundtrack. It creates an ominous sense of menace as the ellipses transform themselves into hyperbolas.


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