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 of this series for the terminology regarding n and (n-1). The result provided below can easily be adapted to a higher number of dimensions, i.e. “n-ellipsoids”.

Surface, shells and volume inside a (n-1)-ellipsoid can be created by a linear transformation of vectors describing a sphere and enclosed spherical shells in the embedding ℝⁿ. The calculation of the ellipsoid’s volume, therefore, requires the proper treatment of a related generating and invertible (positive-definite) (nxn)-matrix M.

In this series, we also look at the stuff of multidimensional volume integrals from the perspective of “Multivariate Normal Distributions” [MVNs]. A natural connection for our present context is provided by the fact that a MVN has ellipsoidal contour surfaces of constant probability density. We will therefore use some of our gathered knowledge about such contour surfaces.

Previous posts

• n-dimensional spheres and ellipsoids – I – Gamma-function, volume and surface of n-spheres

Terms and abbreviations

We work in an Euclidean space represented by the ℝⁿ. We abbreviate a respective Cartesian Coordinate System by CCS. The ellipsoid is described in a system called CCS_E. A related sphere, created by a (back-) transformation of the coordinates, is described in a rotated system CCS_S with scaled axes. We name the n main axes of a general ellipsoid “a_i“. The main axes may be rotated against the coordinate axes of the CCS_E.

Regarding MVNs: The standard deviation of a MVN’s marginal Gaussian function along the i-th coordinate axis is called “σ_i “. A special MVN for a standardized random vector Z is given for independent standardized Gaussians along the coordinate axes of a CCS_S:

We only regard centered distributions, whose points of maximum probability density coincide with the CCS-origin, so μ = 0. Due to the standardization of its constituting independent Gaussians, a Z-distribution has spherical contour surfaces of constant probability density. A general MVN-distribution for a random vector Y, has concentric ellipsoidal contour surfaces, instead. The main axes of the ellipsoids are rotated against the coordinate axes of a general CCS_E. Such a general Y-distribution, i.e. a general MVN, is related to a Z-distribution by a linear invertible transformation matrix M. See this post for a introduction.

Suppositions 1

As we have already determined the volume of a (n-1)-sphere, the derivation of the volume of a (n-1)-ellipsoid is relatively easy. We will make use of the fact that n-ellipsoids (with their center coinciding with the CCS-origin) can be regarded as the result of a two combined linear operations actively applied to position vectors:

• A first operation stretches the axes of the sphere along the coordinate axes and a subsequent one rotates the resulting surface in the ℝⁿ. A spherical shell with an infinitesimal extension dr in radial direction thus becomes an ellipsoidal shell with infinitesimal thickness.

Invertible and positive-definite linear transformations of spherical surfaces define quadratic forms for ellipsoidal surfaces. The above linear operations can be expressed in terms of a invertible and positive-definite (nxn)-matrix M in the ℝⁿ. See a separate section below on the question of how to find such a matrix.

Passive rotation: We can equivalently look at the whole scenario of the creation of an ellipsoid in the ℝⁿ also from the perspective of a proper passive transformation between coordinate systems. Transforming passively would mean: The description of an ellipsoidal surface via vectors expressed in the basis of a chosen CCS_E is equivalent to the description of a spherical surface or shell given by the same vectors, but expressed with respect to the basis of a backward rotated coordinate system CCS_S with properly stretched coordinate axes. Why is this relevant in our context? Answer:
For such a kind of coordinate transformation we know the rules of volume integration in the ℝⁿ very well.

At this point, we could simply analyze the results of an invertible linear transformations applied to vectors that define a unit sphere in the ℝⁿ. But just for fun, we make a small detour. (For those, who like shortcuts: You find a much shorter version of the aspired proof in [2]. )

Relation to MVNs

In this and the following section, I want to motivate a special property of the determinant of the transformation matrix M. It is in general given by a product of the lengths of the target ellipsoid’s semi-axes.

While we could prove this directly by means of linear algebra and standard geometry, we take this insight from properties of a MVN, in particular of its contour surfaces. The transition from spherical contour surfaces of a Z-MVN to ellipsoidal contours of a general Y-MVN can be summarized by the following facts:

1. The contour surfaces of a general MVN’s probability density ρ in the ℝⁿ are given by concentric ellipsoids. Each of these surfaces is a continuous (n-1)-dimensional manifold in the ℝⁿ. The contour surfaces are fully defined by the covariance matrix Σ_Y of the MVN.
2. All vectors on an ellipsoidal contour surface of a MVN result from an invertible linear transformation M applied to the vectors pointing onto a spherical contour surface of a special MVN-distribution for a standardized random vector Z. The probability density of Z is given by the product of n independent, standardized Gaussians (for Z‘s marginals along the coordinate axes).
3. Position vectors to points on an ellipsoidal contour surface of a MVN have a so called constant Mahalanobis distance d_m. The definition of d_m corresponds to a quadratic form of the data points’ coordinates in the chosen CCS_E. One such ellipsoidal surface is that for d_m=1.
4. The determinant of the transformation matrix M is given by a product of the standard deviations σ_i of the MVN’s marginal Gaussian distributions along the coordinate axes.
5. For d_m=1 these standard deviations σ_i just give us the lengths of the semi-axes (= half-axes) a_i of the special contour ellipsoid for d_m=1. (One can show this from an eigendecomposition of covariance matrix Σ=MM^T or its inverse).
6. M itself can be written as a matrix product of a rotation matrix with a diagonal stretching matrix. M = M_rot • M_str.

We have made extensive use of these properties already in a parallel post series on a finite ellipsoidal core of a MVN. See e.g. [2] for details. Regarding BVNs and MVNs, this blog provides a variety of posts for beginners; see e.g. here and related posts.

Suppositions 2

We can write symbolically (knowing that the transformation actually has to be applied to concrete vectors of the distribution):

Regarding the symmetric, invertible and positive definite covariance matrix Σ_Y of the resulting MVN, we have

An ellipsoidal contour surface of the MVN at a Mahalanobis distance d_m=1, is defined by the condition

imposed on respective surface vectors y_e. As said: The eigenvalues of M are given by the square roots σ_i of the variances of the MVN’s n marginal Gaussians.The σ_i just define the squared semi-axes of the contour ellipsoid for d_m=1.

Geometrically speaking: The surface of a contour ellipsoid with semi-axes a_i is just the result of an invertible linear transformation M actively applied to vectors reaching to the surface of a unit sphere by a matrix M within a Cartesian Coordinate System CCS_S. Inversely formulated for a passive transformation of coordinates and vector components:

• We get the ellipsoid by transforming vector components, which define a spherical contour surface in a system CCS_S, into a backward rotated coordinate system CCS_E with axes (!) scaled by 1/σ_i. Note that scaling an axes by 1/σ_i corresponds to stretching semi-axes of the sphere along the coordinate axes by σ_i. A reverse transformation by forward rotating CCS_E and afterward scaling its axes by σ_i (i.e. contracting the semi-axes by 1/σ_i) brings us back to the original system CCS_S where the contour ellipsoid would look spherical. The forward rotation alone aligns the coordinate axes with the main axes of the ellipsoid (rotation to the so called “main axes system”).

For an ellipsoidal contour surface of a MVN with d_m=1 (!) and semi-axes a_i, we know:

It is relatively easy to see, that the latter condition holds in general and that we can abstract from contour ellipsoids of a MVN. We just consider (n-1)-ellipsoids created from (n-1) spheres by an adjusted transformation M. The argument goes as follows:

Let us define the unit sphere and the respective ellipsoid for d_m= 1 by vectors x_s and x_e, respectively

In the middle eq. (6), (d_m)² now just abbreviates the quadratic form “qf” defining the ellipsoid:

In (8) the a_i-values are still specific and reflect the σ_i -values, because we have assume d_m=1. But, note that we can always scale the stretching part of M via scaling the diagonal elements of the matrix M_str for the initial stretching operation by a constant factor! This would simply stretch the ellipsoid’s surface for d_m=1. This means, that we can generalize and abstract to

for any surface with any wished for any a_i-values and any quadratic form. We just have to adjust or choose a suited M. Actually, we do not need references to MVN’s and to Mahalanobis distances any longer. I.e., we can take eq. (4) seriously for any ellipsoid. If this does not convince you, maybe the argumentation in the next section will make you more confident.

Quadratic form and linear transformation of a unit sphere

How can we understand that a quadratic form for a (n-1)-ellipsoid is related to a matrix M_t transforming vectors which define a (n-1)-dimensional unit sphere in the ℝⁿ?

Well, we know the recipe to show this already from other posts (on MVNs 🙂 ). We start with the quadratic form that defines the ellipsoid in a CCS_E via some symmetric (!), invertible, positive definite matrix Σ^-1 (this time, no reference to MVNs required; even ignore the (-1) for the time being, if you like):

The y_e are assumed to be the result of some invertible linear transformation M_t in the ℝⁿ:

If we want the x_s to define a sphere, the chain of matrices must obviously fulfill :

with I_n being the (nxn) identity matrix and λ representing a constant factor. Note that the left side corresponds to a transformation of Σ^-1 into a different coordinate system (=> CCS_S). As M_t is invertible we can rewrite eq. (15) as a condition M_t must fulfill :

If you want, you can now move a factor sqrt(λ) into M_t and M_t^T to get matrices M and M^T. I.e.:

A proper M can always be found by a Cholesky decomposition of Σ. This shows that for a given quadratic form (mediated by some invertible pos.-definite matrix Σ^-1), we can always find a matrix M that creates the ellipsoid from a unit sphere in the ℝⁿ. An eigendecomposition of the symmetric Σ gives us

with Λ being a diagonal matrix having the eigenvalues of Σ on its diagonal. Q instead is an orthogonal matrix. Q can be regarded as a rotation in the ℝⁿ. This shows that M can indeed be written as

As the stretching operation creates the n semi-axis a_i of the ellipsoid, which is only rotated afterward, we can indeed conclude:

As we already found out from the properties of a MVN. For those who have followed my posts on MVNs this is nothing new. But it reflects a relation between MVNs and geometry.

Derivation of the volume of a n-ellipsoid

We need just one more puzzle piece:

• We know that the volume of a body in linearly transformed coordinates can be calculated by performing a volume integral in the original coordinate system, but taking the Jacobian determinant of the coordinate transformation into account.

Let us call the volumes of the unit sphere V^S_n,1 and the volume of our general target ellipsoid V^E_n,_qf. For an invertible coordinate transformation mediated by our matrix M (with constant coefficients), we can relate the volume integrals for the ellipsoid in CCS_E and the sphere in CCS_S:

As said: The first integral applies in the CCS_E of the transformed coordinates, and the 2nd integral in the original coordinate system CCS_S. I.e.,

Using the results of the previous post (see eqs. (38) and (39) there), we get the volume of the transformed unit sphere, i.e. of our (n-1)-ellipsoid:

The (n-1)-ellipsoid and its axes are described in some chosen CCS_E. To get the volume of a n-ellipsoid you just have to adapt the number of dimensions (n -> n+1).

Conclusion / Outlook

The volume of a (n-1)-dimensional ellipsoid in the ℝⁿ can easily be determined when we know the ellipsoids semi-axes (or, equivalently, its quadratic form). We have made the result plausible not only by straightforwardly using Linear Algebra, but also by generalizing properties of contour surfaces of a MVN.

Unfortunately, there is no such derivation of an analytic expression for the surface area of such an ellipsoid. But as we will see in the next post of this series, we may again get help from statistics and MVN-like distributions.

Links and literature

[1] R. Mönchmeyer, 2025, “Covariance matrix of a cut-off Multivariate Normal Distribution – II – integrals over volume and surface of an n-dimensional sphere”, blog post at
https://machine-learning.anracom.com/2025/12/12/covariance-matrix-of-a-cut-off-multivariate-normal-distribution-ii-integrals-over-volume-and-surface-of-an-n-dimensional-sphere/

[2] M.Jorgensen, 2014, “Volumes of n-dimensional spheres and ellipsoids”,
https://www.whitman.edu/Documents/Academics/Mathematics/2014/jorgenmd.pdf