Oblate spheroid

An oblate spheroid is obtained by rotating an ellipse about its shorter axis. If its equatorial semiaxis is one and its polar semiaxis is $q\le 1$, then $q$ is its intrinsic axis ratio.

If the symmetry axis of an oblate spheroid makes angle $i$ with the line of sight, its projected axis ratio satisfiesThus a face-on spheroid appears round and an edge-on spheroid reveals its intrinsic axis ratio.

For an unoriented symmetry axis chosen uniformly on the sphere, $\cos i$ is uniform on $[0,1]$. At fixed intrinsic ratio $q$, the change-of-variables formula for a probability density gives