Dr. S. Shapiro wrote:
> (1) Volume V of sphere = (4/3)*pi*r^3
> (2) Volume V of spheroid = (4/3)*pi*a*b^2
> where axes 2a > 2b
> (3) Eccentricity e of spheroid = [1 - (b^2/a^2)]^(1/2)
> then given a numerical value for the volume of a sphere,
> is there a unique value for the eccentricity of a spheroid
> with the same volume?
Though this might be called an algebra problem, rather than a geometry
one, the answer is No.
There are an infinity of eccentricities of spheroids of a given volume.
Let V be fixed, then solve (2) above for a:
a = 3V/(4*pi*b*b)
Substituting in (3):
e = sqrt[1 - (b*b) * ((4*pi*b*b)^2)/((3*V)^2)]
or, rewriting terms...
e = sqrt(1 - (16/9)*(pi^2)*(b^6)/(V^2))
So e is a continuous function of b for b>=0 and a fixed V.
Peter J. Floriani, Ph.D.
floriani at epix.net
"I have often thanked God for the telephone." G. K. Chesterton, 1910