This is something which I feel intuitively is true but I'm having trouble finding a way of proving it mathematically.
Given an oblate sphere, or ellipsoid, with equation $$x^2+y^2+(z^2 / c^2)=1, c<1$$ i.e. "squashed" only in one direction, will you always be able to measure its major or widest diameter no matter what angle you're viewing it from?
I'm supposing I will need to prove that the 2D projection of the ellipsoid onto a plane perpendicular to the viewing direction preserves the width under any rotational transformations. Does this seem about right? In pursuit of this I'm looking into the matrix representation of the above ellipsoid in order to simplify the projection and transformations (i.e. $\vec v^TA \vec v = 1$ where A is a diagonal matrix with diagonal elements $1, 1, 1/c^2$)
I'm obviously not a mathematician, but this question occurred to me in thinking about navigation in space -- measuring the angular width of a planet can tell you how far away you are from it, but all planets are oblate spheres, bulging at their equator, so can you be certain that no matter what angle you're viewing the planet from you are always able to measure the radius at the equator?
Many thanks all =)
Yes. Consider the part of the spheroid in the $xy$ plane, which is a circle. No matter what angle you view it from, there is a diameter perpendicular to your line of sight, so the maximum extent of the spheroid is at least this. No two points of the spheroid are farther apart, so the maximum is no greater.