Oblate spheroid

An oblate spheroid is the flattened case of a spheroid, with polar semiaxis no greater than its equatorial semiaxis.

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