Somewhat apropos ellipsoid geodesics, On Sat, Apr 24, 2010 at 5:45 PM, Bill Gosper <billgosper@gmail.com> wrote:
In those ubiquitous little tables of area and volume formulae in handbooks and textbooks, there's often a picture of a skewed cone next to the right cone to emphasize that they have the same volume formula. Did you ever wonder why there's never the skewed cone area formula? For the special case with the apex directly above a point on the circumference, I get
pi h r 3F2[-1/2,1/4,3/4; 1/2, 1; -4 r^2/h^2],
which doesn't even converge for h < 2 r.
The area element for the general x,y,z parametric surface is just
Out[85]= (1,0) (0,1) Norm[Cross[({x, y, z}) , ({x, y, z}) ]]
(I don't recall this from 18.03.)
[...] Neil has hanging above his computer a lamellation of not-so-little laminated cards full of those good old geometry formulas, and some not-so-good ones! One clearly states (and illustrates) that the side area of a skew circular cylinder is just s/h times the side area of an unskewed cylinder, where h is the height and s is then slant height! It goes on to claim this is true for *arbitrary* (noncircular) cylinders! For the circular case, radius r, height h, slant height s, Mma and I get 2*r*(s*EllipticE[1 - h^2/s^2] + h*EllipticE[1 - s^2/h^2]). For a slant of 45 degrees, this gives (h*r*(Gamma[-(1/4)]^2 + 4*Gamma[1/4]^2))/(4*Sqrt[2*Pi]) ~7.6404 h r, compared to 2 pi h r for unslanted, and compared to 2 sqrt(2) pi h r ~8.88577 h r for that bogus card! I wonder how many of these cards decorate the walls of Boeing and Lockheed. --rwg