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Geometry problem

Peter J. Floriani, Ph.D. floriani at epix.net
Thu Oct 9 14:30:32 EST 1997

Dr. S. Shapiro wrote:
>      Given
> (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

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