Prolate spheroid ($c>a=b$)

A prolate spheroid is an ellipsoid with two equal shorter semiaxes and a longer symmetry-axis semiaxis. Its meridional section is an ellipse rotated about its major axis. In the coordinates $R^2/(\lambda +\alpha )+z^2/(\lambda +\beta )=1$ with $\beta >\alpha$, a fixed regular $\lambda$ has equatorial radius $\sqrt{\lambda +\alpha }$ and polar radius $\sqrt{\lambda +\beta }$; its focal separation is independent of $\lambda$, giving a confocal family.

For the prolate spheroid $u^2/a^2+(v^2+w^2)/b^2=1$ with $a>b>0$, intersect the concentric sphere $u^2+v^2+w^2=R^2$. When $b<R<a$, subtraction gives two circles in the perpendicular planesTheir centers lie on the long principal axis and the line through the centers is normal to both planes. At $R=b$ the circles merge into the equator; at $R=a$ they shrink to the two polar vertices. There is no intersection for $R<b$ or $R>a$.