Spheroid

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.

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