Further playing with the big hammer dA = dt du |V_t x V_u| gives the area of a 1x1xc spheroid as 2 pi (1 + c^2 asec(c)/sqrt(c^2-1)), with the axaxb general case following easily. This agrees with Mathworld's prolate (c>1) formula, but for oblate, Weisstein gives the logarithmic equivalent that avoids imaginary/imaginary. Something must be wrong with our notation if we need to switch at c = 1. The function there is smooth as a marble!-) A = 2 pi (1 + c 2F1[1,1,3/2,1/2-1/2c]) . (Rotsa ruck coaxing this out of Mma. Or even checking it.) A simpler example might be Plot[ArcCos[x]^2,{x,-1,5}], where the plotter needed to call a complex-valued function to get real results. Should we put acos^2 on pocket calculators? --rwg Joerg: The images are now (again, for me) all visible. Thanks for fixing this. The thanks go to Neil for rescuing the .nb and creating the working .html with his Windows 7 Machine.