ELLIPSOID OF REVOLUTION

 Other name : spheroid.

I) Oblate ellipsoid of revolution.

 Rotation of an ellipse around its minor axis Oz. Cylindrical equation:  with a (half major axis)  (half minor axis) ; cartesian equation: . Cartesian parametrization: , u = latitude, v = longitude. First fundamental quadratic form: . Area element: .  Main radius of curvature: . Gauss curvature:  Area :  where e is the excentricity of the ellipse, area , area of the circumscribed cylindrical box. Volume : .

The oblate ellipsoid of revolution is the surface of revolution obtained by rotating the ellipse around its minor axis, having the shape of a pebble or a flying saucer, or also a go stone.

 Opposite, views of closed geodesics of the oblate spheroid, corresponding to some "turcs head-knots" with 3, 8, 15 crossings. Differential system whose solutions give these geodesics:

II) Prolate ellipsoid of revolution.

 Rotation of an ellipse around its major axis Oz. Cylindrical equation:  with a (half major axis)  (half minor axis). For the other formulas, use those in the previous box by exchanging a and b, except for Aire :  where e is the excentricity of the ellipse, area , area of the circumscribed cylindrical box.

The prolate ellipsoid of revolution is the surface of revolution obtained by rotating the ellipse around its major axis, having a shape of a
cigar or rugby ball.

 Opposite, views of four closed geodesics of the prolate spheroid. With above-below crossings, the second gives a figure eight knot, the third gives a knot with 9 crossings 9.1.40, and the fourth a knot with 12 crossings 12.1.1019.

Compare these geodesics with these of the torus.