On Fri, 16 Mar 2001, Sarah Edmondson-Jones wrote: > Ramanujan came up with an approximation: > > ( 3(a + b) - sqrt[ (a + 3b)(3a + b) ] ) pi > > for major and minor half axes a and b.
Curiouser and curiouser! I found another, simpler, approximation on a geometry site: 2 * Pi * sqrt((a^2 +b^2)/2) Both being approximations, it's not surprising to find that they agree for a circle (a = b), but disagree more and more at large eccentricities. What I found interesting was that for extreme eccentricities, where the minor axis approaches zero relative to the major axis, Ramanujan's estimate evaluates to less than the straight-line approximation, while the other estimate evaluates to greater. (As the ellipse collapses to nearly a straight line, it seems clear that the perimeter approaches that of a rhombus, 4 * sqrt(a^2 + b^2)...) Off to look for more "guesses"! Dave Bell 37.3N121.9W
