Dear Richard, Just between you and me: Here's my method for constructing a nice baseball curve: First divide a sphere S into 2 hemispheres by drawing an equator Q. Now draw 2 great semicircles C_1, C_2 on S, with their endpoints on Q -- at -pi/2, 0, pi/2, pi -- with their midpoints at the poles. Conformal mapping theory (e.g., in Ahlfors, 3rd ed.) tells us that S minus C_1 u C_2 is conformally equivalent to a right cylinder Cyl of fixed ratio (of diameter to height). And the equivalence is unique up to a rotation. Call the conformal equivalence F: F: S - (C_1 u C_2) -> Cyl The cylinder has a well-defined "waist" circle W (halfway between its boundary circles). Finally, take the inverse image B of W by F: B := Finv(W) and then B is the baseball curve. (I mentioned this on math-fun to Conway maybe 15 years ago, and he objected because C_1 and C_2 are part of a continuum of curves that we could also use instead: curves with the same symmetry with respect to S, but longer -- or maybe even shorter -- portions of a great circle.) Every so often I work on figuring out the exact formula for B -- which I expect to be simple -- but I haven't done that yet. Regards, Dan On 2013-11-03, at 9:09 AM, rkg wrote:
Dear funsters, A tennis-ball appears to be made from 2 congruent pieces of material, seamed together in a curve. Are possible equations to the curve known? I'd like a smooth algebraic equation, probably of degree 4, and preferably with a maximum number of rational points on it. A first approximation might be to take a sphere of radius root(3) and centre at (0,0,0) and take the 8 great circle arcs (1,1,1) to (-1,1,1) to (-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), (1,1,-1) and back to (1,1,1). However, this isn't smooth at the 8 corners of the cube, and I think that it doesn't even partition the sphere into two congruent pieces.
Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun