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n-spheres and n-ellipsoids

orthogonal projections of ellipsoid

Orthogonal projections of multidimensional ellipsoids – I – points on the ellipsoid that give us the surface points of the projection

In the mathematical section of this blog we deal with the geometry of Multivariate Normal Distributions [MVNs]. We found that the contour surfaces of MVNs are multidimensional ellipsoids given by quadratic forms of their constituting vectors. We have identified the variance-covariance matrix of a MVN as the inverse matrix mediating the required quadratic form. We also considered ellipsoidal cores of… Read More »Orthogonal projections of multidimensional ellipsoids – I – points on the ellipsoid that give us the surface points of the projection

3-dim projections of 4-dim MVN

n-dimensional spheres and ellipsoids – IV – numerical check of Rivin’s formula for the surface areas of ellipsoids in 3 dimensions and the perimeters of ellipses

In the third post of this series we have discussed an idea of I. Rivin (see [1], [2]). Rivin has shown that the surface area of a general (n-1)-dimensional ellipsoid in a n-dimensional Euclidean space can be expressed in terms of an expectation value of a specific function weighted by a multivariate Gaussian probability density [pdf]. In contrast to (n-1)-spheres… Read More »n-dimensional spheres and ellipsoids – IV – numerical check of Rivin’s formula for the surface areas of ellipsoids in 3 dimensions and the perimeters of ellipses

n-dimensional spheres and ellipsoids – III – Surface area of n-dimensional ellipsoid and its relation to MVN-statistics

In the 2nd post of this series we have derived an explicit formula for the volume of a n-dimensional ellipsoid (in an Euclidean space). One reason for the relatively simple derivation was that the determinant of the generating linear transformation could be taken in front of the volume integral. Unfortunately, an analogue sequence of steps is not possible for an… Read More »n-dimensional spheres and ellipsoids – III – Surface area of n-dimensional ellipsoid and its relation to MVN-statistics

Concentric surfaces of ellipsoids

n-dimensional spheres and ellipsoids – II – volume of n-ellipsoids

This post series provides some insights into the nature of n-dimensional spheres and ellipsoids and the derivation of some special integrals over their volumes. In this 2nd post we look at the volume of a “(n-1)–ellipsoid“. This term refers to a closed ellipsoidal surface in a n-dimensional Euclidean space. Such a surface is a (n-1)-dimensional manifold. See the 1st post… Read More »n-dimensional spheres and ellipsoids – II – volume of n-ellipsoids