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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.

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
headknots" 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 abovebelow 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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© Robert FERRÉOL 2021