Word Finder - ellipsoids

definition

A surface, all of whose cross sections are elliptic or circular (including the sphere), that generalises the ellipse and in Cartesian coordinates (x, y, z) is a quadric with equation x2/a2 + y2/b2 + z2/c2 = 0.

definition

Such a surface used as a model of the shape of the earth.

example

Here the geoid is thirty meters below the ellipsoid.

Legendre, in 1783, extended Maclaurin's theorem concerning ellipsoids of revolution to the case of any spheroid of revolution where the attracted point, instead of being limited to the axis or equator, occupied any position in space; and Laplace, in his treatise Theorie du mouvement et de la figure elliptique des planetes (published in 1784), effected a still further generalization by proving, what had been suspected by Legendre, that the theorem was equally true for any confocal ellipsoids.

The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity U and 15 1 of equation (II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular area a l.

The potential of such a shell at any internal point is constant, and the equi-potential surfaces for external space are ellipsoids confocal with the ellipsoidal shell.

Legendre shows that Maclaurin's theorem with respect to confocal ellipsoids is true for any position of the external point when the ellipsoids are solids of revolution.

During forty years the resources of analysis, even in the hands of d'Alembert, Lagrange and Laplace, had not carried the theory of the attraction of ellipsoids beyond the point which the geometry of Maclaurin had reached.

The third memoir relates to Laplace's theorem respecting confocal ellipsoids.

The problem is identical with that of finding the common conjugate diameters of the ellipsoids T(x, y, I) =const., V(x, y, 1) =const.

For example, all ellipsoids referred to co-ordinates parallel to any three conjugate diameters are parallel projections of each other and of a sphere referred to rectangular co-ordinates.

Among his most remarkable works may be mentioned his ten memoirs on quantics, commenced in 1854 and completed in 1878; his creation of the theory of matrices; his researches on the theory of groups; his memoir on abstract geometry, a subject which he created; his introduction into geometry of the "absolute"; his researches on the higher singularities of curves and surfaces; the classification of cubic curves; additions to the theories of rational transformation and correspondence; the theory of the twenty-seven lines that lie on a cubic surface; the theory of elliptic functions; the attraction of ellipsoids; the British Association Reports, 1857 and 1862, on recent progress in general and special theoretical dynamics, and on the secular acceleration of the moon's mean motion.

Add the error ellipsoids of each shot to get an error ellipsoid for the entire loop.

If the structure was refined anisotropically, the orientations and the magnitudes of vibrational ellipsoids should be displayed.