Total Ellipse of the Heart
Here's an ellipse for you.
Play with the focus A. (Its twin moves automatically.)
These two foci belong to a whole bunch of ellipses, but point B selects just one of them.
Play with point B.
The line through B and the ellipse's center splits the ellipse in half. The slider flips one of those halves.
Play with the slider.
Motivated by Howie Hua.
How was this made?
I often use GeoGebra to build interactive visual demonstrations of mathematical ideas. The mathematics required to make one of these gadgets is usually more complicated than the mathematics being demonstrated, and that's certainly the case here.
I'm not even sure I had a mathematical demonstration in mind, beyond the ideas that
- An ellipse is a curve that is a little bit like a circle, and its construction has something to do with "foci";
- If you cut an ellipse the right way and flip one half over, you sometimes get a heart (the icon, not the organ--wouldn't THAT be neat!).
- Given the foci of an ellipse centered at the origin, transform that ellipse into the unit circle centered at the origin. (Also see where a given point on that ellipse goes, so I can relate any half of the ellipse to the corresponding half of the circle.)
- Given half of the unit circle, transform it back to fill the corresponding half of the original ellipse.
- Repeat #2, but include a parameter (controlled by the slider) in order to smoothly transition from exactly #2 to a version of #2 that is reflected across a line perpendicular to the indicated line.
- Open a desktop installation of GeoGebra.
- Download my applet (or any GeoGebra applet, for that matter) from the Details screen (off the drop-down menu from the three dots in the top right corner, I think).
- Open the downloaded .ggb file in GeoGebra.
- Explore however you like, but I especially recommend that you use the Construction Protocol view to see the parts in the order in which they were built.