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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.
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.
 
Opposite, views of a closed geodesic of the oblate ellipsoid, whose top view forms a crossed octahedron.

 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.
 
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© Robert FERRÉOL 2020