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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:
Solid semi-ellipsoid of revolution around z-axis a = the radius of the base circle h = the height of the semi-ellipsoid from the base cicle's center to the edge
The ellipsoid method generates a sequence of ellipsoids whose volume uniformly decreases at every step, thus enclosing a minimizer of a convex function. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a number of steps that is ...
The prolate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis c and semi-minor axis a; therefore, e may again be identified as the eccentricity. (See ellipse.) [3] These formulas are identical in the sense that the formula for S oblate can be used to calculate the surface area of a prolate spheroid and vice ...
An alternative parametrization exists that closely follows the angular parametrization of spherical coordinates: [1] = , = , = . Here, > parametrizes the concentric ellipsoids around the origin and [,] and [,] are the usual polar and azimuthal angles of spherical coordinates, respectively.
A sphere (top), rotational ellipsoid (left) and triaxial ellipsoid (right) The volume of a sphere of radius R is 4 3 π R 3 {\displaystyle {\frac {4}{3}}\pi R^{3}} . Given the volume of a non-spherical object V , one can calculate its volume-equivalent radius by setting
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution respectively. Other terms used are ellipticity , or oblateness . The usual notation for flattening is f {\displaystyle f} and its definition in terms of the semi-axes a {\displaystyle a} and b {\displaystyle b} of ...
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.