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 Incoming links: Oblate spheroid

Choose intrinsic Cartesian coordinates in which the oblate spheroid is

X^{2} + Y^{2} + \frac{Z ^{2}}{q ^{2}} = 1.

Let

ℓ

be distance along the line of sight and let

y

be the sky coordinate in the plane containing the line of sight and the symmetry axis. A rotation through the inclination

i

gives

Y = y cos i - ℓ sin i, Z = y sin i + ℓ cos i .

Substitution into the ellipsoid equation and minimization over

ℓ

, equivalently requiring the quadratic in

ℓ

to have zero discriminant on the projected boundary, gives

X^{2} + \frac{y ^{2}}{c o s ^{2} i + q ^{2} s i n ^{2} i} = 1.

Q^{2} = cos^{2} i + q^{2} sin^{2} i .

It correctly gives

Q = 1

face-on and

Q = q

edge-on.

i

with the line of sight, its projected axis ratio satisfies

Q^{2} = cos^{2} i + q^{2} sin^{2} i .

Thus a face-on spheroid appears round and an edge-on spheroid reveals its intrinsic axis ratio.