On 2/16/07, David W. Cantrell <DWCantrell@sigmaxi.net> wrote:
Hmm. Maybe not. :-( Anyway, the figure at <http://img255.imageshack.us/img255/498/eggconstructzn0.gif> should make the construction abundantly clear. I suspect that that beautiful egg shape was used a good bit during the Middle Ages (perhaps in architecture?) but have no supporting evidence.
I looked at this diagram, afterwards attempted to reconstruct it mentally, and failed --- in spite of the fact that it has much in common with the construction of a Reuleaux (constant diameter) polygon.
Having been a maker of historical instruments, I analyzed the outlines of many lutes (and some other instruments) from the Renaissance. The outlines were always done, it seemed, by ruler-and-compass constructions. But the outlines of lutes in the later Renaissance involved somewhat more complicated constructions than did Arnault's lute.
A highly polished surface reflecting a bright light would show up the discontinuous curvature of such an elementary spline; as technology advanced, so perfectionist makers might well have become dissatisfied with this situation --- as have the designers of modern cars. Fred Lunnon