Spherical Stochastic Geometry (SSG)

• Spherical Stochastic Geometry is a framework studying random geometric structures on spheres with features like rotational invariance, geodesic metrics, and curvature-dependent corrections.
• It models processes including tessellations, random inscribed polytopes, and Poisson point processes to derive explicit formulas for network and geometric metrics.
• SSG bridges theoretical and applied research by enabling precise analysis of communication networks, geometric inequalities, and interference with unique spherical effects.

Spherical stochastic geometry (SSG) studies random geometric structures on the sphere $S^d\subset \mathbb R^{d+1}$ or on concentric spherical shells. In the literature considered here, it includes random tessellations generated by great-sphere cuts, random inscribed polytopes, Poisson hyperplane and Voronoi cells in spherical space, and communication-network point processes on $S^2(R)$. Its defining features are rotational invariance, geodesic or chordal distance, spherical caps as extremal bodies, and curvature-dependent corrections that have no Euclidean analogue; in several tessellation problems, compactness replaces the Euclidean large-cell regime by a high-intensity regime (Deuß et al., 2015, Hug et al., 2017, Wang et al., 13 Jan 2025).

1. Geometric setting and probabilistic primitives

A basic object in the geometric branch of SSG is the spherically convex body

$K=S^d\cap C,$

where $C\subset\mathbb R^{d+1}$ is a closed convex cone containing no line. The family of such bodies is denoted $\mathcal K$, with proper subclass $\mathcal K_g$. Standard size functionals include the spherical volume $V_s(K)=\sigma_d(K)$, spherical surface area $A_s(K)$, and spherical intrinsic volumes $V_i(K)$. Centered inradius and circumradius are

$$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$

and a basic deviation functional is $S^2(R)$0 (Hug et al., 2017).

On a sphere of radius $S^2(R)$1, a homogeneous Poisson point process (PPP) $S^2(R)$2 with intensity $S^2(R)$3 satisfies

$S^2(R)$4

for any measurable surface patch $S^2(R)$5, and

$S^2(R)$6

For a uniformly distributed point on the sphere, the central-angle distribution relative to a fixed pole has

$S^2(R)$7

with cap CDF $S^2(R)$8 (Wang et al., 13 Jan 2025).

In communication-oriented SSG, great-circle geometry is often explicit. For $S^2(R)$9,

$K=S^d\cap C,$0

and in spherical coordinates

$K=S^d\cap C,$1

Path-loss models then use $K=S^d\cap C,$2 in place of Euclidean distance, for example

$K=S^d\cap C,$3

(Wang et al., 21 Jul 2025).

The NTN survey classifies spherical SG models into three types based on orbital modeling methods: non-orbital models, stochastic-orbital models, and fixed-orbital models. Concrete examples include homogeneous PPP and binomial point process (BPP) models on spherical shells, Cox PP and Dual Stochastic BPP constructions for random orbits, and Poisson Line Process and Orbit Geometry Model formulations for fixed orbital structure (Wang et al., 13 Jan 2025). This taxonomy indicates that SSG is not restricted to a single point-process family; rather, it provides a common spherical state space in which isotropy, visibility, and curvature are handled directly.

2. Great-circle tessellations and spherical STIT analogues

Deuß, Hörrmann, and Thäle introduced a random recursive cell splitting scheme on the $K=S^d\cap C,$4-dimensional unit sphere as the spherical analogue of the STIT tessellation process. Let $K=S^d\cap C,$5 be the set of all great circles, equipped with the rotation-invariant probability measure $K=S^d\cap C,$6. Starting from the equator-split tessellation

$K=S^d\cap C,$7

each cell $K=S^d\cap C,$8 waits an exponential time with rate $K=S^d\cap C,$9, where $C\subset\mathbb R^{d+1}$0, and is then split by a random great circle chosen from $C\subset\mathbb R^{d+1}$1 according to the conditional law $C\subset\mathbb R^{d+1}$2. The generator is

$C\subset\mathbb R^{d+1}$3

and a corresponding martingale identity allows first-order moment calculations (Deuß et al., 2015).

The induced tessellation admits explicit formulas for total edge, maximal-segment, and side lengths: $C\subset\mathbb R^{d+1}$4 For the intensities,

$C\subset\mathbb R^{d+1}$5

$C\subset\mathbb R^{d+1}$6

Typical-object mean sizes are likewise explicit, including

$C\subset\mathbb R^{d+1}$7

$C\subset\mathbb R^{d+1}$8

(Deuß et al., 2015).

The same model yields a capacity functional. For a closed connected set $C\subset\mathbb R^{d+1}$9 lying entirely in one open hemisphere,

$\mathcal K$0

For a union of disjoint connected sets $\mathcal K$1 in one hemisphere, the avoidance probability satisfies a recursive integral formula involving $\mathcal K$2 and the separation sets $\mathcal K$3 (Deuß et al., 2015).

A notable structural result concerns intersections with a fixed great circle $\mathcal K$4. The point processes

$\mathcal K$5

are independent, and each is a Poisson point process on the corresponding open semi-circle with intensity $\mathcal K$6 (Deuß et al., 2015). This gives an exact one-dimensional trace of the spherical splitting process.

Comparison with Euclidean STIT makes the curvature effect transparent. The polynomial term $\mathcal K$7 survives as the dominant asymptotic, but the spherical formulas contain genuinely curved-space corrections such as $\mathcal K$8 and $\mathcal K$9. In particular, the term $\mathcal K_g$0 in $\mathcal K_g$1 and the extra $\mathcal K_g$2 in $\mathcal K_g$3 have no flat-plane analogue; as $\mathcal K_g$4, these exponential terms vanish and the leading polynomial behavior matches the Euclidean case (Deuß et al., 2015).

3. Random inscribed polytopes on $\mathcal K_g$5

Another major branch of SSG studies the convex hull of random points on the unit sphere in $\mathcal K_g$6. In the uniform model, $\mathcal K_g$7 points are chosen independently and uniformly on $\mathcal K_g$8, and their convex hull is $\mathcal K_g$9. In the Poisson model, $V_s(K)=\sigma_d(K)$0 is the convex hull of a stationary Poisson point process of intensity $V_s(K)=\sigma_d(K)$1 on $V_s(K)=\sigma_d(K)$2, equivalently obtained by first drawing $V_s(K)=\sigma_d(K)$3 and then choosing $V_s(K)=\sigma_d(K)$4 i.i.d. uniform points on $V_s(K)=\sigma_d(K)$5. Because all sample points lie on $V_s(K)=\sigma_d(K)$6, none is interior, so $V_s(K)=\sigma_d(K)$7 almost surely and $V_s(K)=\sigma_d(K)$8 (Akopyan et al., 2020).

A central result is that a uniformly chosen facet $V_s(K)=\sigma_d(K)$9 of $A_s(K)$0 or $A_s(K)$1 is acute with probability $A_s(K)$2. The argument uses a spherical analogue of a Blaschke–Petkantschin decomposition together with an Archimedes-style cap-area computation showing that, for three random uniformly chosen points on $A_s(K)$3, circumcircle radius is independent of shape and a random Euclidean triangle on $A_s(K)$4 is acute with probability $A_s(K)$5 (Akopyan et al., 2020).

The expected intrinsic volumes of the random hull are explicit. With $A_s(K)$6 denoting half the total edge-length and $A_s(K)$7 surface area, the uniform model satisfies

$A_s(K)$8

where $A_s(K)$9 and $V_i(K)$0. In addition,

$V_i(K)$1

For the Poisson model, conditioning on $V_i(K)$2 leads to formulas involving modified Bessel functions $V_i(K)$3, for example

$V_i(K)$4

with analogous expressions for $V_i(K)$5 and $V_i(K)$6 (Akopyan et al., 2020).

The expected total edge length also has closed forms: $V_i(K)$7 and

$V_i(K)$8

The paper further derives the distribution of the minimum Euclidean distance $V_i(K)$9 from a fixed north pole $$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$0 to the hull: $$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$1 and for the geodesic distance $$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$2,

$$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$3

(Akopyan et al., 2020).

These formulas extend to an ellipsoid $$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$4 equipped with the homeoid density, defined as the push-forward of the uniform measure on $$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$5. In that setting, volume ratios are preserved, area and width inherit the same rational prefactors as on the sphere, and

$$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$6

(Akopyan et al., 2020). The extension shows that several spherical exact formulas are covariant under invertible linear maps once the sampling density is adjusted appropriately.

4. Geometric inequalities, stability, and high-intensity asymptotics

Hug and Reichenbacher developed a spherical analogue of the geometric-inequality approach to Kendall’s problem. A central hitting functional is

$$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$7

with $$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$8. In the Voronoi setting, another functional is

$$r_s(K)=\max_{e\in K} r_e(K), \qquad R_s(K)=\min_{e\in K} R_e(K),$$9

For a Poisson hyperplane tessellation of $S^2(R)$00 driven by an isotropic Poisson point process of intensity $S^2(R)$01, the Crofton cell $S^2(R)$02 is the a.s. unique cell containing a fixed reference point, while the typical cell $S^2(R)$03 is defined via a rotation-invariant marking and Palm-distribution argument. Their expectations are linked by

$S^2(R)$04

for rotation-invariant measurable $S^2(R)$05 (Hug et al., 2017).

The spherical Urysohn inequality states that if $S^2(R)$06 and $S^2(R)$07 is a spherical cap with $S^2(R)$08, then

$S^2(R)$09

with equality if and only if $S^2(R)$10 is a cap. A corresponding inequality holds for $S^2(R)$11 in the Voronoi setting. Quantitative strengthenings relate excess hitting functional to deviation from a cap. In particular, if the comparison cap radius is at most $S^2(R)$12, then

$S^2(R)$13

where $S^2(R)$14 is the $S^2(R)$15-deviation of the radial function; and if $S^2(R)$16 and $S^2(R)$17, then

$S^2(R)$18

(Hug et al., 2017).

These deterministic inequalities feed directly into probabilistic deviation bounds. For suitable size functional $S^2(R)$19 and deviation functional $S^2(R)$20, there are constants $S^2(R)$21 such that

$S^2(R)$22

and the same form holds for the typical cell $S^2(R)$23. In the Voronoi case,

$S^2(R)$24

(Hug et al., 2017).

The asymptotic regime is fundamentally different from the Euclidean one. For the Crofton cell and the typical cell,

$S^2(R)$25

where

$S^2(R)$26

Because $S^2(R)$27 is compact, there is no analogue of the Euclidean large-cell regime; the natural asymptotic is $S^2(R)$28, and the resulting limit laws are exponential tails rather than Gaussian fluctuations (Hug et al., 2017). A common misconception is that spherical cell asymptotics should mirror planar central-limit normalization. The spherical theory shows that compactness changes the regime itself.

5. SSG in non-terrestrial networks and the planar-versus-spherical question

The NTN literature presents SSG as a recently proposed analytical framework that has garnered increasing attention because it is suitable for modeling large-scale dynamic topologies and provides an analytical framework for interference analysis and low-complexity performance evaluation. The survey of NTN models organizes spherical SG into non-orbital, stochastic-orbital, and fixed-orbital classes, and introduces topology-level concepts such as association strategy, central angle, zenith angle, contact angle, and availability probability, together with channel-level models for large-scale fading, small-scale fading, and beam gain (Wang et al., 13 Jan 2025).

A central methodological issue is whether planar SG is adequate or whether Earth’s curvature must be retained. Wang et al. formalize this by introducing a point process generation algorithm that simultaneously generates a pair of homogeneous and asymptotically similar planar and spherical point processes. For each matched pair, one draws $S^2(R)$29, sets

$S^2(R)$30

for the spherical cap point, and

$S^2(R)$31

for the planar point. If

$S^2(R)$32

then, as $S^2(R)$33 with fixed cap area, each spherical point flattens out and coincides with its planar match. The paper then defines topology-related metrics such as Earth Mover’s Distance, Wasserstein-1, and contact distance CDF, network-level metrics such as average SINR, coverage probability, and average rate, and the relative error

$S^2(R)$34

(Wang et al., 21 Jul 2025).

This framework also yields an analytical expression for the optimal planar altitude for the Wasserstein-1 metric: $S^2(R)$35 The reported case studies are sharply scale-dependent. For $S^2(R)$36, such as HAPs, $S^2(R)$37 under typical metrics; for $S^2(R)$38, such as LEO, $S^2(R)$39 unless extremely narrow beams are used. A specific HAP case with $S^2(R)$40 and $S^2(R)$41 gives $S^2(R)$42, whereas a LEO case with $S^2(R)$43 and $S^2(R)$44 gives $S^2(R)$45 (Wang et al., 21 Jul 2025).

The NTN survey reaches a closely related conclusion from matched BPP case studies. Mapping a spherical cap of half-angle $S^2(R)$46 to a planar disk and comparing user-satellite distances yields $S^2(R)$47 for a HAP at $S^2(R)$48 within LoS, $S^2(R)$49 for a LEO at $S^2(R)$50 within a narrow beam $S^2(R)$51, and a requirement of $S^2(R)$52 beam for MEO at $S^2(R)$53 (Wang et al., 13 Jan 2025). Taken together, these results do not reject planar models outright; rather, they localize the regime in which planar approximation is quantitatively defensible.

6. Relay, patrol, and energy-aware extensions

Recent application papers extend SSG from single-layer coverage models to structured and mobility-aware aerial and satellite systems. In a spherical-stochastic-geometry analysis of SAGIN from the relay perspective, the point-process layout consists of ground users as a PPP $S^2(R)$54 of density $S^2(R)$55 on $S^2(R)$56, HAPs on $S^2(R)$57, and satellites as a BPP of $S^2(R)$58 points on $S^2(R)$59. The paper introduces three performance metrics: the average access data rate, the average backhaul data rate, and the backhaul rate exceedance probability (BREP). It derives analytical expressions for these quantities, including a closed-form expression for the end-to-end performance metric BREP, under assumptions that RF fading is approximated by Gamma and FSO fading is purely pointing-error, with low-SINR approximations used in the rate expressions (Tarhouni et al., 27 Jul 2025).

The same paper also gives exact spherical distance laws that drive the rate analysis. For example, the angular distance between two independent uniform points on $S^2(R)$60 has density

$S^2(R)$61

and, when conditioned on visibility $S^2(R)$62,

$S^2(R)$63

The authors emphasize several SSG-specific features: curvature and global coverage, rotational invariance and tractable integrals, exact visibility bounds on interferers, and satellite-constellation effects that enter through the backhaul-distance distribution (Tarhouni et al., 27 Jul 2025).

A further development is the patrol-based HAP framework on the spherical Earth. This model introduces two small-circle ring Cox processes on the shell $S^2(R)$64: the small-circle ring Poisson Cox process (SCR-PCP), in which each patrol ring carries a one-dimensional PPP, and the small-circle ring binomial Cox process (SCR-BCP), in which each ring contains exactly $S^2(R)$65 uniformly distributed platforms. Both constructions are isotropic under rotations. The paper derives the nearest-anchor, nearest-ring, and nearest-HAPs distance distributions, together with the joint serving distance and serving ring angle distribution required for SCR-BCP analysis, then obtains coverage probability under nearest-HAPs association by decomposing aggregate interference into same-ring and other-ring components and characterizing their conditional Laplace transforms (Shah et al., 4 Jun 2026).

The patrol model also incorporates steady circular flight propulsion and defines the coverage energy efficiency metric

$S^2(R)$66

where

$S^2(R)$67

An interior optimum satisfies

$S^2(R)$68

yielding an analytical condition for the energy-optimal patrol radius (Shah et al., 4 Jun 2026). This introduces a distinctive SSG theme absent from classical planar models: geometry, interference, and propulsion are coupled through the same spherical state space.

The NTN survey lists several advanced topics that remain only partly explored, including multi-shell extensions, mobility and time-varying topology, non-uniform satellite distributions, routing on random spherical graphs, physical-layer security, satellite clustering, energy harvesting, and satellite-enabled positioning (Wang et al., 13 Jan 2025). This suggests that current SSG research is expanding from isotropic shell models toward models with orbital structure, temporal dynamics, and engineering constraints, while retaining the same core probabilistic language of spherical point processes, visibility regions, and curvature-dependent performance functionals.