---
title: Projected Isotropic Normal Distribution
url: https://www.emergentmind.com/topics/projected-isotropic-normal-distribution
type: topic
---

# Projected Isotropic Normal Distribution

Searching arXiv for recent papers on the projected isotropic normal distribution and closely related projected normal models.
The projected isotropic normal distribution is the distribution obtained by projecting an isotropic Gaussian random vector onto a lower-dimensional linear subspace or onto a unit sphere, circle, or hypertorus, depending on the projection map. In the Euclidean linear-projection setting, an isotropic Gaussian $X \sim \mathcal{N}(0,\sigma^2 I_d)$ remains Gaussian under every orthogonal projection, so any fixed one-dimensional projection is exactly $\mathcal{N}(0,\sigma^2)$ and any $k$-dimensional orthogonal projection is $\mathcal{N}(0,\sigma^2 I_k)$ [1109.2227, 2502.02397]. In the directional setting, the normalized variable $Y=X/\|X\|$ lies on the unit sphere and follows a projected normal, also called an angular Gaussian; under isotropy and zero mean this reduces to the uniform law on the sphere, whereas nonzero mean induces a symmetric, unimodal directional law determined by the ratio of mean magnitude to isotropic noise scale [2506.17461, 2410.22384, 2603.03816]. Recent work also studies random linear modulation $Y_n=\Xi_n'X_n$, where isotropic normal inputs and spherically symmetric modulators generate exact or asymptotic normal variance-mixtures, together with quantitative convergence, matrix-normal limits, and characterizations of when the modulator itself must be Gaussian [2510.12928].

## 1. Euclidean definition and isotropic projection principle

In its most basic form, the projected isotropic normal begins with an isotropic Gaussian random vector
$$
X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).
$$
The isotropy condition means that the covariance is $\sigma^2 I_d$, so the law is rotationally invariant around its mean when $\mu=0$ [1109.2227, 2502.02397]. If $P\in\mathbb{R}^{k\times d}$ has orthonormal rows, $PP^\top=I_k$, and one defines the linear projection
$$
y = Px,
$$
then
$$
y \sim \mathcal{N}_k(P\mu, P\Sigma P^\top).
$$
In the isotropic case $\Sigma=\sigma^2 I_d$, this simplifies to
$$
P\Sigma P^\top=\sigma^2 I_k,
$$
hence every linear projection is again isotropic normal in the projected subspace [2502.02397].

Several immediate consequences are standard. For any unit vector $u$,
$$
u^\top X \sim \mathcal{N}(u^\top\mu,\sigma^2),
$$
and in the zero-mean isotropic case,
$$
u^\top X \sim \mathcal{N}(0,\sigma^2),
$$
identical for every unit-length direction by isotropy [2510.12928, 1109.2227]. Likewise, an orthogonal projection onto any $k$-dimensional subspace yields $\mathcal{N}(0,\sigma^2 I_k)$ on that image subspace [1109.2227].

This Euclidean notion should be distinguished from directional projection. In linear projection, the image remains a Gaussian distribution in $\mathbb{R}^k$; in spherical or circular projection, the vector is normalized by its norm and the image lies on a manifold such as $\mathbb{S}^{d-1}$ or $\mathbb{S}^1$ [2506.17461, 2410.22384]. The shared terminology reflects the common geometric origin in isotropic Gaussian structure.

## 2. Radial–angular structure and projection to spheres

For $X\sim\mathcal{N}(0,\sigma^2 I_d)$, the radial–angular decomposition is especially simple. Writing $R=\|X\|$ and $U=X/\|X\|$ with $U\in\mathbb{S}^{d-1}$, one has that $R$ and $U$ are independent, and $U$ is uniform on $\mathbb{S}^{d-1}$ [1109.2227]. The radius satisfies
$$
\frac{R^2}{\sigma^2}\sim\chi^2_d,
$$
and its density is the chi density
$$
f_R(r)=\frac{1}{\sigma^d 2^{\frac d2-1}\Gamma(d/2)}\,r^{d-1}\exp\!\Big(-\frac{r^2}{2\sigma^2}\Big),\quad r>0
$$
[1109.2227].

From this viewpoint, isotropy means that the entire directional distribution is encoded by the radial law. When the mean vanishes, projecting onto the unit sphere produces no directional preference: the projected normal is exactly uniform [1109.2227, 2506.17461]. When the mean is nonzero, the projected distribution becomes concentrated around the direction of the mean, but the isotropic covariance still enforces axisymmetry around that direction [2410.22384, 2603.03816].

A constructive 2D limit mechanism is given by the radial central limit theorem. Let $\boldsymbol{X}$ be supported on a line through the origin in the plane, with zero mean and unit variance, and define
$$
\boldsymbol{Z}_N=\frac{1}{\sqrt N}\big(\mathcal{R}_{\theta_1}\boldsymbol{X}_1+\cdots+\mathcal{R}_{\theta_N}\boldsymbol{X}_N\big),
$$
with equally spaced angles $\theta_k=(k-1)\pi/N$. Then $\boldsymbol{Z}_N$ converges in distribution to the standard normal distribution on the plane [1109.2227]. This yields a probabilistic interpretation of isotropic Gaussian structure from rotated one-dimensional components, and it clarifies why angular homogenization plus averaging produces a radial Gaussian limit.

In high dimensions, the isotropic Gaussian radius exhibits thin-shell concentration. Since $R^2/\sigma^2\sim\chi_d^2$ with mean $d$ and variance $2d$, the radius concentrates around $\sigma\sqrt d$, with $O(1)$ fluctuations rather than $O(\sqrt d)$ fluctuations [1109.2227]. This thin-shell behavior is directly connected to the high-dimensional projection limits studied for random modulation and conditional normality [2510.12928].

## 3. Directional projected isotropic normal on the circle and sphere

On the circle, the projected isotropic normal arises from
$$
X=(X_1,X_2)^\top\sim\mathcal{N}_2(\mu,\sigma^2 I_2),
$$
with phase angle
$$
\Theta=\operatorname{atan2}(X_2,X_1).
$$
Writing $\mu=\rho(\cos\phi,\sin\phi)^\top$, the distribution depends on the mean direction $\phi$ and the scalar concentration parameter
$$
\gamma=\rho^2/(4\sigma^2),
$$
equivalently $\mathrm{SNR}=2\gamma=\rho^2/(2\sigma^2)$ [2603.03816]. Its exact density is
$$
f_\Theta(\theta;\phi,\gamma)=\frac{1}{2\pi}e^{-2\gamma}
+2\sqrt{\gamma}\cos(\theta-\phi)\Phi(2\sqrt{\gamma}\cos(\theta-\phi))\varphi(2\sqrt{\gamma}\sin(\theta-\phi)),
$$
for $\theta\in[-\pi,\pi]$, where $\varphi$ and $\Phi$ are the standard normal pdf and cdf [2603.03816]. Under isotropy, the density depends only on $\theta-\phi$, is unimodal with mode at $\theta=\phi$, and is symmetric:
$$
f_\Theta(\phi+t;\gamma)=f_\Theta(\phi-t;\gamma)
$$
[2603.03816].

An equivalent isotropic-circle parametrization uses $\kappa=\|\mu\|/\sigma$ and $\alpha=\operatorname{atan2}(\mu_2,\mu_1)$, giving
$$
f(\theta)=\frac{e^{-\kappa^2/2}}{2\pi}
+\frac{\kappa\cos(\theta-\alpha)}{\sqrt{2\pi}}
e^{-(\kappa^2/2)\sin^2(\theta-\alpha)}
\Phi(\kappa\cos(\theta-\alpha))
$$
[1711.10463]. The two formulations are consistent through $\gamma=\rho^2/(4\sigma^2)$.

On $\mathbb{S}^2$, if $X\sim N(\mu,\sigma^2 I_3)$ and
$$
Y=P(X)=X/\|X\|,
$$
then $Y$ has a projected normal distribution on the sphere [2410.22384]. Its density with respect to surface area measure can be written as
$$
f_Y(y)=\int_0^\infty (2\pi\sigma^2)^{-3/2}\exp\!\big(-\|ry-\mu\|^2/(2\sigma^2)\big)r^2\,dr,
$$
and, in closed form,
$$
f_Y(y)=(2\pi)^{-3/2}\exp\!\big(-\|\mu\|^2/(2\sigma^2)\big)\left[D+(D^2+1)\frac{\Phi(D)}{\varphi(D)}\right],
$$
with
$$
D=(y\cdot \mu)/\sigma
$$
[2410.22384]. As $\sigma\to\infty$, this tends to the uniform density $1/(4\pi)$ on $\mathbb{S}^2$ [2410.22384].

A more general sphere-valued projected normal is defined by
$$
y=x/\|x\|,\qquad x\sim\mathcal{N}(\mu,\Sigma),
$$
with density on $\mathbb{S}^{n-1}$ expressed through
$$
q_1=\mu^\top\Sigma^{-1}\mu,\quad q_2=\mu^\top\Sigma^{-1}y,\quad q_3=y^\top\Sigma^{-1}y,
$$
and a recursively defined function $\mathcal{M}_{n-1}$ [2506.17461]. In the isotropic specialization $\Sigma=\sigma^2 I$, these simplify to $q_1=\|\mu\|^2/\sigma^2$, $q_2=(\mu^\top y)/\sigma^2$, and $q_3=1/\sigma^2$, and when $\mu=0$ the law becomes uniform on $\mathbb{S}^{n-1}$ [2506.17461].

A common misconception is to identify the projected isotropic normal with the von Mises or von Mises–Fisher family. The cited works explicitly distinguish them: the projected isotropic normal arises from Euclidean Gaussian normalization, while the von Mises and von Mises–Fisher families are exponential-family models on compact manifolds [2410.22384, 2508.16432, 2603.03816]. Approximation links exist, but the models are not the same.

## 4. Exact moments, intrinsic statistics, and identifiability

For the sphere-valued isotropic projected normal $y=x/\|x\|$ with $x\sim\mathcal{N}(\mu,\sigma^2 I)$ in $\mathbb{R}^n$, exact isotropic moment formulas are available. The mean has the form
$$
\bar y = a\,\mu,
$$
where
$$
a=\frac{\Gamma(n/2+1/2)}{\sqrt{2\sigma^2}\,\Gamma(n/2+1)}\,
{}_1F_1\!\left(\frac12;\frac{n+2}{2};-\frac{\|\mu\|^2}{2\sigma^2}\right)
$$
[2506.17461]. The second moment is
$$
\mathbb{E}[yy^\top]
=
\frac{1}{n}\,
{}_1F_1\!\left(1;\frac n2+1;-\frac{\|\mu\|^2}{2\sigma^2}\right)I
+
\frac{1}{n+2}\,
{}_1F_1\!\left(1;\frac n2+2;-\frac{\|\mu\|^2}{2\sigma^2}\right)\mu\mu^\top,
$$
and the covariance can be written as
$$
\Psi=b\,\mu\mu^\top+c\,I
$$
with coefficients $b$ and $c$ given explicitly in terms of confluent hypergeometric functions [2506.17461]. When $\mu=0$, these formulas reduce to $\bar y=0$ and $\mathbb{E}[yy^\top]=(1/n)I$, the uniform case [2506.17461].

On $\mathbb{S}^2$, intrinsic statistics provide a geometric formulation. If $Y=P(X)$ with $X\sim N(\mu,\sigma^2 I_3)$, the intrinsic Fréchet mean is
$$
m=\arg\min_{z\in S^2}\mathbb{E}[d_{S^2}(z,Y)^2],
$$
and under isotropy the expectation commutes with projection:
$$
E_{\mathrm{intr}}[Y]=P(E[X])=\mu/\|\mu\|=:m,
$$
provided $\mu\neq 0$ [2410.22384]. If $\mu=0$, the distribution is uniform on $S^2$ and the intrinsic mean is not unique [2410.22384].

The intrinsic covariance on the tangent plane $T_mS^2$ is isotropic:
$$
\Sigma_{S^2}=f(\lambda)I_2,\qquad \lambda=\sigma^2/\|\mu\|^2,
$$
where $f$ is a continuous, strictly increasing bijection from $\lambda\in[0,\infty)$ onto $v\in[0,(\pi^2-4)/4)$ [2410.22384]. This yields a one-to-one correspondence between the scale-free Euclidean parameter $\lambda$ and the intrinsic covariance of the projected spherical distribution [2410.22384].

Identifiability is a central issue. Phase-only or direction-only data do not identify $\|\mu\|$ and $\sigma^2$ separately; only their ratio is identifiable. On the circle, the identifiable parameters are $\phi$ and $\gamma=\rho^2/(4\sigma^2)$ [2603.03816]. On $\mathbb{S}^2$, the estimable quantities from projected data alone are the direction $m=\mu/\|\mu\|$ and the scale-free variance $\lambda=\sigma^2/\|\mu\|^2$ [2410.22384]. More generally, the projected normal inherits a scale non-identifiability: scaling a Gaussian pair $(W_{i1},W_{i2})$ by a positive constant leaves the angle unchanged [1711.10463]. Bayesian treatments therefore impose identification constraints such as fixing one variance per circular pair and post-processing posterior samples accordingly [1711.10463].

For the circular projected isotropic normal, recent work derives exact trigonometric moments. If $\Theta\sim \mathrm{PIN}(\phi,\gamma)$, then
$$
m_k=\mathbb{E}[e^{ik\Theta}]=\rho_k e^{ik\phi},\qquad \mathbb{E}[\sin(k(\Theta-\phi))]=0,
$$
and, for integer $p\ge 1$,
$$
\mathbb{E}[\cos(p(\Theta-\phi))]
=
\sqrt{\pi\gamma/2}\,e^{-\gamma}\{I_{(p-1)/2}(\gamma)+I_{(p+1)/2}(\gamma)\}
$$
[2603.03816]. In particular,
$$
\mathbb{E}[\cos(\Theta-\phi)]
=
\sqrt{\pi\gamma/2}\,e^{-\gamma}\{I_0(\gamma)+I_1(\gamma)\},
$$
and
$$
\mathbb{E}[\cos 2(\Theta-\phi)] = 1-(e^{-\gamma}/\gamma)\sinh\gamma
$$
[2603.03816]. These closed forms make the mean resultant and its square analytically accessible.

## 5. Random projections, variance-mixtures, and high-dimensional modulation

A distinct but closely related regime considers scalar projections formed by random modulation:
$$
Y_n=\Xi_n'X_n,
$$
where $X_n,\Xi_n\in\mathbb{R}^{d_n}$ are independent [2510.12928]. When
$$
X_n\sim \mathcal{N}_{d_n}(\mu_n,\sigma_n^2 I_{d_n}),
$$
the conditional law is exact:
$$
Y_n\mid \Xi_n=\xi \sim \mathcal{N}(\xi'\mu_n,\sigma_n^2\|\xi\|^2).
$$
In the zero-mean isotropic case,
$$
Y_n\mid \Xi_n \sim \mathcal{N}(0,\sigma_n^2\|\Xi_n\|^2)
$$
[2510.12928].

Thus the unconditional law is a normal variance-mixture with mixing variable
$$
S_n=\sigma_n^2\|\Xi_n\|^2.
$$
Its density and characteristic function are
$$
f_{Y_n}(y)=\mathbb{E}\Big[(2\pi S_n)^{-1/2}\exp\{-y^2/(2S_n)\}\Big],
$$
$$
\phi_{Y_n}(t)=\mathbb{E}\Big[\exp\{-(1/2)S_n t^2\}\Big]
$$
[2510.12928]. This statement is exact for isotropic Gaussian $X_n$ and independent spherically symmetric $\Xi_n$.

If $\Xi_n$ is itself Gaussian, $\Xi_n\sim\mathcal{N}_{d_n}(0,I_{d_n})$, and $X_n$ satisfies the thin-shell and zero-overlap conditions (C.1) and (C.2), then
$$
Y_n\mid \Xi_n \xrightarrow{w} \mathcal{N}_1(0,\sigma^2)
$$
as $n\to\infty$, and the limit does not depend on the realized $\Xi_n$ [2510.12928]. More generally, if $X_{n,1},\dots,X_{n,k}$ and $\Xi_{n,1},\dots,\Xi_{n,l}$ are i.i.d. copies in the Gaussian-modulator setting and
$$
Y_{n;j,r}=\Xi_{n,j}'X_{n,r},
$$
then the $l\times k$ matrix $Y_n=(Y_{n;j,r})$ satisfies
$$
Y_n\mid(\Xi_{n,1},\dots,\Xi_{n,l}) \xrightarrow{w} Z,
$$
where $Z$ has independent entries $Z_{j,r}\sim\mathcal{N}_1(0,\sigma^2)$; equivalently,
$$
\mathrm{vec}(Y_n)\mid(\Xi_{n,\cdot}) \xrightarrow{w} \mathcal{N}_{lk}(0,\sigma^2 I_{lk}),
$$
that is, a matrix normal limit $MN_{l\times k}(0;\sigma^2 I_l,I_k)$ [2510.12928].

When $\Xi_n$ is only spherically symmetric, Schoenberg’s characterization yields a Gaussian scale-mixture representation at the characteristic-function level:
$$
\psi(t)=\int_0^\infty \exp(-tv^2/2)\,dG(v),\qquad t\ge 0,
$$
for some distribution function $G$ on $[0,\infty)$ [2510.12928]. This leads to pointwise and uniform convergence of conditional densities and distribution functions to mixtures of centered normal laws $\mathcal{N}_1(0,\sigma^2V^2)$ under assumptions (C.1)–(C.3) and integrability conditions [2510.12928].

The quantitative rate is governed by the Gram-matrix deviation term
$$
\mathbb{E}\|A_{n,j}-\sigma^2I_j\|_F^2
=
j\,\mathbb{E}(\|X_n\|^2-\sigma^2)^2
+
j(j-1)\,[\mathbb{E}(X_n'\widetilde X_n)]^2,
$$
which separates thin-shell variance from zero-overlap concentration [2510.12928]. This suggests that the asymptotic normality of random isotropic projections is controlled by the extent to which the sample cloud becomes radially concentrated and mutually orthogonal in high dimension.

## 6. Characterization results, approximations, and statistical inference

A sharp characterization of Gaussian modulators is obtained through Pólya’s theorem. Suppose $X_n$ satisfies (C.1)–(C.2), is independent of $\Xi_n$, and $\Xi_n$ is spherically symmetric with characteristic function $\psi_0(\|u\|^2)$. Then
$$
\Xi_n\sim\mathcal{N}_{d_n}(0,\sigma_0^2 I_{d_n})
$$
for some $\sigma_0$ if and only if, for all $t\in\mathbb{R}$,
$$
\mathrm{Var}_{\Xi_n}(\phi_{Y_n\mid \Xi_n}(t))\to 0
$$
as $n\to\infty$ [2510.12928]. Equivalently,
$$
\psi_0(t^2)=\big[\psi_0(2^{-1}t^2)\big]^2,
$$
which is Pólya’s characterization and forces normality [2510.12928]. The paper also gives a counterexample using a spherically symmetric $\alpha$-stable modulator, for which the variance condition fails [2510.12928].

For the circle-valued projected isotropic normal, the exact sampling distribution of the mean resultant is analytically intricate, so approximations are built from the von Mises resultant law [2603.03816]. Two mappings $\kappa(\gamma)$ are proposed. The first is moment matching:
$$
A(\kappa)=I_1(\kappa)/I_0(\kappa)
=
\mathbb{E}_{\mathrm{PIN}}[\cos(\Theta-\phi)]
=
\sqrt{\pi\gamma/2}\,e^{-\gamma}\{I_0(\gamma)+I_1(\gamma)\},
$$
so $\kappa=A^{-1}(\cdot)$ [2603.03816]. The second is score matching:
$$
\kappa
=
\frac{2\,\mathbb{E}_{\mathrm{PIN}}[\cos(\Theta-\phi)]}
{1-\mathbb{E}_{\mathrm{PIN}}[\cos(2(\Theta-\phi))]}
=
\sqrt{2\pi\gamma}\{I_0(\gamma)+I_1(\gamma)\}\cdot (\gamma/\sinh\gamma)
$$
[2603.03816]. Both satisfy $\kappa\approx \sqrt{2\pi\gamma}$ for small $\gamma$ and $\kappa\approx 4\gamma$ for large $\gamma$ [2603.03816].

Inference for the circular model is based on the likelihood
$$
\ell(\phi,\gamma)=\sum_{i=1}^n \log f_\Theta(\theta_i;\phi,\gamma),
$$
with numerical maximization [2603.03816]. Moment-based estimation sets $\hat\phi$ equal to the sample mean direction and solves
$$
\bar R=\sqrt{\pi\gamma/2}\,e^{-\gamma}\{I_0(\gamma)+I_1(\gamma)\}
$$
for $\gamma$ [2603.03816]. Under uniformity, the Rayleigh statistic satisfies
$$
2n\bar R^2 \overset{a}{\sim}\chi_2^2
$$
for large $n$ [2603.03816].

On $\mathbb{S}^2$, parameter estimation proceeds geometrically: compute the sample Fréchet mean $\hat m$, estimate the intrinsic covariance $\hat\Sigma_{S^2}$, set $\hat v=\mathrm{trace}(\hat\Sigma_{S^2})/2$, and invert the bijection $v=f(\lambda)$ numerically to recover $\hat\lambda$ [2410.22384]. Because only $m$ and $\lambda$ are identifiable from spherical data alone, this is a scale-free estimation procedure unless external information fixes either $\|\mu\|$ or $\sigma^2$ [2410.22384].

For the general sphere-valued projected normal, moment approximations based on Taylor expansions and quadratic-form identities yield analytic approximations to $\mathbb{E}[y]$ and $\mathbb{E}[yy^\top]$ for $y=x/\|x\|$ with $x\sim\mathcal{N}(\mu,\Sigma)$ [2506.17461]. The same paper develops moment-matching estimation using the objective
$$
(1-\lambda)\|\tilde y-\bar y_{\mathrm{obs}}\|^2 + \lambda\|\tilde\Psi-\Psi_{\mathrm{obs}}\|_F^2,
$$
with manifold constraints on $\mu$ and $\Sigma$ [2506.17461]. In the isotropic case, these methods can be constrained to $\Sigma=\sigma^2 I$ [2506.17461].

## 7. Related families, applications, and extensions

The projected isotropic normal appears in several distinct application domains. In neuroscience, a recent phase-analysis model for EEG under flash stimulation assumes
$$
Y_j=\mu+e_{1j}+ie_{2j},
$$
where $e_{1j},e_{2j}$ are independent Gaussian with equal variance. The resulting phase $\Theta_j=\arg(Y_j)$ has the projected isotropic normal distribution on the circle [2603.03816]. The paper studies the mean resultant, the component synchrony measure $\bar R^2$, approximation-based tests of phase locking, and an EEG application in which electrode O1 under 6 Hz flashing showed $\mathrm{CSM}\approx 0.9939$ and a hybrid estimate $\hat\gamma_{\mathrm{HYB}}\approx 41.24$ [2603.03816].

In multivariate visualization, an isotropic reference normal $N_d(\mu,\sigma^2 I_d)$ projects to circular contours in every 2D view:
$$
\|y-P\mu\|_2=\sigma c,
$$
where $c^2$ is the original $d$-dimensional Mahalanobis threshold [2502.02397]. This geometry underlies a projection-pursuit index for anomaly detection relative to a multivariate normal baseline, implemented in the R package `tourr` [2502.02397].

In directional and toroidal statistics, isotropic projected normal structure also underlies symmetric, unimodal models on the circle and hypertorus. For the univariate circle case, if $(X_c,X_s)^\top\sim N_2((\eta_c,\eta_s)^\top,\Psi)$ with $\Psi=I_2$, the projected normal is symmetric around its mean direction $\mu=\operatorname{atan}^*(\eta_s/\eta_c)$ and its concentration depends only on $\kappa=\sqrt{\eta_c^2+\eta_s^2}$ [2508.16432]. The corresponding toroidal projected normal family $TPN_d(\mu,\kappa,\Sigma)$ is closed under marginalization, and each univariate marginal is a symmetric, unimodal projected normal on the circle [2508.16432]. This family supports Bayesian inference via latent radii and Metropolis–Hastings or Gibbs-type updates [2508.16432].

A broader poly-cylindrical extension combines projected normals for circular components with skew-normal linear components, yielding the joint projected normal and skew-normal distribution [1711.10463]. That model preserves closure under marginalization, highlights the inherited scale non-identifiability of projected normals, and resolves it by fixing one variance per circular pair and post-processing posterior draws [1711.10463]. The isotropic circular case appears as the specialization $\Sigma_{w,i}=\sigma_i^2 I_2$ within this larger construction [1711.10463].

Finally, recent generalizations replace normalization by $\|x\|$ with
$$
\sqrt{x^\top Bx+c},
$$
leading to projected distributions on ellipsoids or inside ellipsoids [2506.17461]. These include $\mathcal{PN}_c(\mu,\Sigma,c)$ on the unit ball and $\mathcal{PN}_{Bc}(\mu,\Sigma,B,c)$ on an ellipsoid interior, with moment approximations and density formulas obtained by linearization and change of variables [2506.17461]. This suggests that the projected isotropic normal is best viewed not as an isolated model, but as the isotropic core of a larger class of normalized Gaussian constructions spanning Euclidean projection, directional statistics, random modulation, and manifold-valued inference.

Source: https://www.emergentmind.com/topics/projected-isotropic-normal-distribution