Grave’s confocal ellipses

Drawing an ellipse with a thread provides a way to construct a confocal ellipses and “gives” curious facts about the life of these curves.

By taking a thread with tied ends which enclosing foci and stretching it with a pencil, draw an ellipse. Threads of different lengths give a whole family of confocal ellipses. But it is possible to enclose one of the already obtained ellipses instead of the segment connecting the foci.

Having drawn the ellipse, let’s make it’s “hard” copy out of a thick enough mate­rial, and then align the copy with the orig­inal. Then we take a thread loop, which encloses the ellipse, and, pulling it with a pencil, draw an oval line. It turns out that this line is also an ellipse, confocal with the orig­inal one. The state­ment can be veri­fied exper­i­men­tally by choosing the length of the loop that encloses the foci.

This theorem was proved in the XIX century by the Charles Graves, an Irish bishop and math­e­mati­cian. In Felix Klein’s book “Lec­tures on higher geom­etry” (German: Vorlesungen Über Höhere Geome­trie) one of the para­graphs is called “Thread construc­tions by Graves and Staude” (German: Fadenkon­struk­tionen von Graves und Staude).

The proof of Graves’ theorem by methods of differ­en­tial geom­etry can be found in Klein’s book. Note that it is not elemen­tary, the reason is that at each moment the thread as a line consists of two segments tangent to the ellipse and an arc of the ellipse. Surpris­ingly, the arc of an ellipse is a very complex object, its length is defined by a complex formula (so-called elliptic inte­grals are being used). And all we did was squeezing a circle, of which every­thing is known!

The “phys­ical” proof of Graves’ theorem, based on the intu­itive asser­tion that the tension forces of straight sections of a thread are equal, is known. But whether there is a proof at the level of elemen­tary geom­etry — without the use of limits and deriv­a­tives — is still unknown, and many geome­ters are still trying to find it.