pm_ellipsoid Module Reference

This module contains classes and procedures for setting up and computing the properties of the hyper-ellipsoids in arbitrary dimensions. More...

Data Types
interface	getCountMemberEll
	Generate and return the number of points that are members (i.e., inside) of the specified \(\ndim\)-dimensional ellipsoid. More...
interface	getLogVolEll
	Generate and return the natural logarithm of the volume of an \(\ndim\)-dimensional ellipsoid. More...
interface	getLogVolUnitBall
	Generate and return the natural logarithm of the volume of an \(\ndim\)-dimensional ball of unit-radius. More...
interface	getVolUnitBall
	Generate and return the natural logarithm of the volume of an \(\ndim\)-dimensional ball of unit-radius. More...
interface	isMemberEll
	Generate and return .true. if and only if the input point is a member (i.e., inside) of the specified \(\ndim\)-dimensional ellipsoid. More...
interface	setLogVolUnitBall
	Return the natural logarithm of the volume of an \(\ndim\)-dimensional ball of unit-radius. More...
interface	setVolUnitBall
	Return the volume of an \(\ndim\)-dimensional ball of unit-radius. More...

Variables
character(*, SK), parameter	MODULE_NAME = "@pm_ellipsoid"

Detailed Description

This module contains classes and procedures for setting up and computing the properties of the hyper-ellipsoids in arbitrary dimensions.

An ellipsoid in Euclidean geomtery \(\ell\) is defined by its representative Gramian matrix \(\Sigma_\ell\), containing all points in \(\mathbb{R}^\ndim\) that satisfy,

\begin{equation} \large (X - \mu_\ell)^T ~ \Sigma_\ell^{-1} ~ (X - \mu_\ell) \leq 1 ~, \end{equation}

where \(\mu_\ell\) represents the center of the ellipsoid, \((X - \mu_\ell)^T\) is the transpose of the vector \((X - \mu_\ell)\), and \(\Sigma_\ell^{-1}\) is the inverse of the matrix \(\Sigma_\ell\).

The volume of this ellipsoid is given by,

\begin{equation} \large V(\ell) = V_\ndim \sqrt{\left| \Sigma_\ell \right|} ~, \end{equation}

where \(\left|\Sigma_\ell\right|\) is the determinant of \(\Sigma_\ell\) and,

\begin{equation} \large V_\ndim = \frac{\pi^{\ndim / 2}}{\up\Gamma(1 + \ndim / 2)} = \begin{cases} \frac{1}{(\ndim/2)!} \pi^{\ndim/2} & \text{if $\ndim$ is even} \\\\ 2^\ndim \frac{1}{\ndim!} \big( \frac{\ndim-1}{2} \big)! ~ \pi^{(\ndim-1)/2} & \text{if $\ndim$ is odd} \end{cases} \end{equation}

is the volume of an \(\ndim\)-ball (that is, a unit-radius \(\ndim\)-dimensional hyper-ball).
It is readily seen that the corresponding unit-volume ellipsoid \(\widehat\ell\) has the representative Gramian matrix,

\begin{equation} \large \Sigma_{\widehat\ell} = V_\ell^{-2/\ndim} ~ \Sigma_\ell ~. \end{equation}

More generally, to scale an ellipsoid \(\ell\) by some factor \(\alpha\) along each coordinate axis, it suffices to be used the new scaled ellipsoid \(\ell^*\) with the representative Gramian matrix,

\begin{equation} \large \Sigma_{\ell^*} = \alpha^2 ~ \Sigma_\ell ~, \end{equation}

in which case, the volume of \(\ell^*\) becomes,

\begin{equation} \large V_{\ell^*} = \alpha^\ndim ~ V_\ell ~. \end{equation}

The surface area of an \(\ndim\)-ball
The surface area \(S\) of an \(\ndim\)-dimensional ball of unit-radius is related to its volume \(V\) as,

\begin{equation} S_\ndim = \ndim \times V_\ndim ~, \end{equation}

where the natural logarithm of \(V\) is returned by getLogVolUnitBall and setLogVolUnitBall.

Note

1. An ellipsoid with a diagonal representative Gramian matrix \(\Sigma_\ell\) is an uncorrelated hyper-ellipsoid whose axes are parallel to the coordinate axes.
   
2. An uncorrelated hyper-ellipsoid with equal diagonal elements in its representative Gramian matrix \(\Sigma_\ell\) is a hyper-ball.
   

See also

pm_distUnifEll
pm_distUnifPar
pm_distMultiNorm
n-sphere
n-ball

Test:

test_pm_ellipsoid

Todo:

Critical Priority: The excluded procedure getLogVolUnion() in this module needs cleanup and merging with this module.

Final Remarks ⛓

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This software is distributed under the MIT license with additional terms outlined below.

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Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, April 23, 2017, 1:36 AM, Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin

Variable Documentation

MODULE_NAME

character(*, SK), parameter pm_ellipsoid::MODULE_NAME = "@pm_ellipsoid"

Definition at line 111 of file pm_ellipsoid.F90.