Bill G: These theorems are in the style of Tristan Needham's "Visual Complex Analysis": http://usf.usfca.edu/vca I fiddled with incenter/inradius with complex numbers, but the formulae weren't so pretty. Since the incenter lies on the angle bisectors, we need sqrt's, which can get very messy. Perhaps you were more successful? Polygon area, polygon-point winding # are just discrete versions of the usual complex plane integrations, right? I'm still intrigued by the Bell Labs paper: Lagarias, Mallows, Wilks. "Beyond the Descartes Circle Theorem". AMM 109 (2002), 338-361. arXiv math.MG/0101066. This paper generalizes an old circle packing theorem to complex numbers. Very cool! See also Graham, et al, "Apollonian Circle Packings:I,II,III,..." arXiv ... At 06:44 PM 6/4/2012, Gosper wrote:
Henry: <minimum publishable quantum, fun facts> These are more than fun facts--they're valuable optimizations.
We should gather up our various complex plane computational geometry slick tricks (circumcenter, circumradius, incenter, inradius, polygon area, polygon-point winding number, line-segment intersection, ...) and publish them somehow.
Henry: (This is also true in the arts; take a look at Beethoven's manuscripts -- not everyone wrote music like Mozart, which were perfect the first time.)
This is probably because Mozart wrote almost nothing in C#. Even less in C++.