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The volume of an ellipsoid is 2 / 3 the volume of a circumscribed elliptic cylinder, and π / 6 the volume of the circumscribed box. The volumes of the inscribed and circumscribed boxes are respectively:
This is a list of volume formulas of basic shapes: [4]: 405–406 ... Ellipsoid – , where , , and are ...
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, ... If A = 2a is the equatorial diameter, and C = 2c is the polar diameter, the volume is
An ellipse is defined by two axes: the major axis (the longest diameter) of length and the minor axis (the shortest diameter) of length , where the quantities and are the lengths of the semi-major and semi-minor axes respectively.
The latter is close to the mean sea level, and therefore an ideal Earth ellipsoid has the same volume as the geoid. While the mean Earth ellipsoid is the ideal basis of global geodesy, for regional networks a so-called reference ellipsoid may be the better choice. [1]
Outer Löwner–John ellipsoid containing a set of a points in . In mathematics, the John ellipsoid or Löwner–John ellipsoid E(K) associated to a convex body K in n-dimensional Euclidean space can refer to the n-dimensional ellipsoid of maximal volume contained within K or the ellipsoid of minimal volume that contains K.
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system (,,) that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces , the ellipsoidal coordinate system is based on confocal quadrics .
Before Jacobi, the Maclaurin spheroid, which was formulated in 1742, was considered to be the only type of ellipsoid which can be in equilibrium. [2] [3] Lagrange in 1811 [4] considered the possibility of a tri-axial ellipsoid being in equilibrium, but concluded that the two equatorial axes of the ellipsoid must be equal, leading back to the solution of Maclaurin spheroid.