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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 |

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 |

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