--- title: Spherical Stochastic Geometry (SSG) url: https://www.emergentmind.com/topics/spherical-stochastic-geometry-ssg type: topic --- # Spherical Stochastic Geometry (SSG) 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 [1511.04232][1709.06522][2503.13455]. ## 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 \(\Delta(K)=R_s(K)-r_s(K)\) [1709.06522]. On a sphere of radius \(R\), a homogeneous Poisson point process (PPP) \(\Phi\) with intensity \(\lambda\) satisfies \[ {\mathbb P}\{\Phi(\mathcal A)=n\} =\frac{(\lambda |\mathcal A|)^n}{n!}\exp(-\lambda |\mathcal A|), \qquad n=0,1,2,\dots, \] for any measurable surface patch \(\mathcal A\subset\mathcal S\), and \[ \mathbb E\{\Phi(\mathcal S)\}=\lambda(4\pi R^2). \] For a uniformly distributed point on the sphere, the central-angle distribution relative to a fixed pole has \[ f_\Phi(\phi)=\frac{\sin\phi}{2},\qquad 0\le \phi\le \pi, \] with cap CDF \(\mathbb P\{\Phi\le \phi\}=(1-\cos\phi)/2\) [2503.13455]. In communication-oriented SSG, great-circle geometry is often explicit. For \(x,y\in S^2(R)\), \[ d_{gc}(x,y)=R\arccos\!\biggl(\frac{x\cdot y}{R^2}\biggr), \] and in spherical coordinates \[ d_{gc}=R\arccos\bigl(\sin\theta_1\sin\theta_2\cos(\phi_1-\phi_2)+\cos\theta_1\cos\theta_2\bigr). \] Path-loss models then use \(d_{gc}\) in place of Euclidean distance, for example \[ \ell(d_{gc})=(d_{gc})^{-\alpha},\qquad \alpha>2 \] [2508.00010]. 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 [2503.13455]. 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 \(2\)-dimensional unit sphere as the spherical analogue of the STIT tessellation process. Let \(G(S^2)\) be the set of all great circles, equipped with the rotation-invariant probability measure \(\tau\). Starting from the equator-split tessellation \[ Y_0=\{S^2_+,S^2_-\}, \] each cell \(p\in Z(Y_t)\) waits an exponential time with rate \(\tau([p])\), where \([p]=\{g\in G(S^2): p\cap g\neq\emptyset\}\), and is then split by a random great circle chosen from \([p]\) according to the conditional law \(\tau(\cdot\cap[p])/\tau([p])\). The generator is \[ L f(T)=\sum_{p\in Z(T)} \int_{[p]} \bigl[f(T\oslash_{p,g})-f(T)\bigr]\tau(dg), \] and a corresponding martingale identity allows first-order moment calculations [1511.04232]. The induced tessellation admits explicit formulas for total edge, maximal-segment, and side lengths: \[ L_E(t)=2\pi[1+t-e^{-2t}],\qquad L_M(t)=2\pi t,\qquad L_S(t)=4\pi(1+t). \] For the intensities, \[ \lambda_Z(t)=t^2+2t+2,\qquad \lambda_E(t)=3(t^2+2t),\qquad \lambda_V(t)=2(t^2+2t), \] \[ \lambda_M(t)=t^2+2t,\qquad \lambda_S(t)=4(t^2+2t)+2e^{-t}. \] Typical-object mean sizes are likewise explicit, including \[ a_Z(t)=\frac{4\pi}{t^2+2t+2},\qquad \ell_{\partial Z}(t)=\frac{4\pi(1+t)}{t^2+2t+2}, \] \[ \ell_E(t)=\frac{2\pi(1+t-e^{-2t})}{3(t^2+2t)},\qquad \ell_M(t)=\frac{2\pi}{t+2} \] [1511.04232]. The same model yields a capacity functional. For a closed connected set \(K\subset S^2\) lying entirely in one open hemisphere, \[ 1-T_{Y_t}(K)=\mathbb P(Y_t\cap K=\emptyset)=\exp[-t\,\tau([K])]. \] For a union of disjoint connected sets \(K=K_1\cup\cdots\cup K_m\) in one hemisphere, the avoidance probability satisfies a recursive integral formula involving \(\tau([{\rm conv}\,K])\) and the separation sets \([Z_1|Z_2]\) [1511.04232]. A notable structural result concerns intersections with a fixed great circle \(g_0\neq A\). The point processes \[ Y_t\cap (S^2_+\setminus A)\cap g_0 \quad\text{and}\quad Y_t\cap (S^2_-\setminus A)\cap g_0 \] are independent, and each is a Poisson point process on the corresponding open semi-circle with intensity \(t\cdot[\text{normalized }\sigma_1\text{-length}]\) [1511.04232]. This gives an exact one-dimensional trace of the spherical splitting process. Comparison with Euclidean STIT makes the curvature effect transparent. The polynomial term \(t^2+2t\) survives as the dominant asymptotic, but the spherical formulas contain genuinely curved-space corrections such as \(e^{-t}\) and \(e^{-2t}\). In particular, the term \(e^{-2t}\) in \(L_E(t)\) and the extra \(2e^{-t}\) in \(\lambda_S(t)\) have no flat-plane analogue; as \(t\to\infty\), these exponential terms vanish and the leading polynomial behavior matches the Euclidean case [1511.04232]. ## 3. Random inscribed polytopes on \(S^2\) Another major branch of SSG studies the convex hull of random points on the unit sphere in \(\mathbb R^3\). In the uniform model, \(n\) points are chosen independently and uniformly on \(S^2\), and their convex hull is \(X_n\). In the Poisson model, \(X_\lambda\) is the convex hull of a stationary Poisson point process of intensity \(\lambda\) on \(S^2\), equivalently obtained by first drawing \(N\sim{\rm Poisson}(4\pi\lambda)\) and then choosing \(N\) i.i.d. uniform points on \(S^2\). Because all sample points lie on \(S^2\), none is interior, so \(V_0(X_n)=n\) almost surely and \(\mathbb E[V_0(X_\lambda)]=4\pi\lambda\) [2007.07783]. A central result is that a uniformly chosen facet \(F\) of \(X_n\) or \(X_\lambda\) is acute with probability \(1/2\). 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 \(S^2\), circumcircle radius is independent of shape and a random Euclidean triangle on \(S^2\) is acute with probability \(1/2\) [2007.07783]. The expected intrinsic volumes of the random hull are explicit. With \(V_1\) denoting half the total edge-length and \(V_2\) surface area, the uniform model satisfies \[ \mathbb E[V_1(X_n)] = V_1(B^3)\cdot \frac{n-1}{n+1}, \qquad \mathbb E[V_2(X_n)] = {\rm Area}(B^3)\cdot \frac{n-1}{n+1}\cdot\frac{n-2}{n+2}, \] where \(V_1(B^3)=2\) and \({\rm Area}(B^3)=4\pi\). In addition, \[ \mathbb E[V_3(X_n)]={\rm Vol}(B^3)\cdot \frac{n-1}{n+1}\cdot\frac{n-2}{n+2}\cdot\frac{n-3}{n+3}. \] For the Poisson model, conditioning on \(N\) leads to formulas involving modified Bessel functions \(I_\alpha\), for example \[ \mathbb E[V_1(X_\lambda)] =2\pi \lambda^{1/2} e^{-2\pi\lambda} I_{1.5}(2\pi\lambda)\cdot V_1(B^3), \] with analogous expressions for \(V_2\) and \(V_3\) [2007.07783]. The expected total edge length also has closed forms: \[ \mathbb E[L(X_n)] =\binom{n}{3}\frac{512}{3\pi}B(n-\tfrac12,\tfrac52), \] and \[ \mathbb E[L(X_\lambda)] =\frac{128}{3}\lambda^{1/2}\bigl(2\pi\lambda^{1/2}e^{-2\pi\lambda}I_2(2\pi\lambda)\bigr). \] The paper further derives the distribution of the minimum Euclidean distance \(R_n\) from a fixed north pole \(N\) to the hull: \[ \mathbb P(R_n\ge r)=\left(1-\frac{r^2}{4}\right)^n, \] and for the geodesic distance \(\Phi_n\), \[ \mathbb P(\Phi_n\ge \phi)=\cos^{2n}(\phi/2) \] [2007.07783]. These formulas extend to an ellipsoid \(\Lambda(S^2)=\partial E\) equipped with the homeoid density, defined as the push-forward of the uniform measure on \(S^2\). In that setting, volume ratios are preserved, area and width inherit the same rational prefactors as on the sphere, and \[ \mathbb E[L({\rm hull}_n)] = \frac12\,{\rm Width}(E)\cdot \mathbb E[L(X_n)] \] [2007.07783]. 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 \[ U_1(K)=\frac12\int_{S^d}\mathbf 1\{x^\perp\cap K\neq \emptyset\}\,d\omega_d(x), \] with \(\Phi(K)=2U_1(K)\). In the Voronoi setting, another functional is \[ \Psi(K)=\int_{S^d}\mathbf 1\{(x-0)^\perp\cap K\neq\emptyset\}\,d\omega_d(x). \] For a Poisson hyperplane tessellation of \(S^d\) driven by an isotropic Poisson point process of intensity \(\gamma\,\omega_d\), the Crofton cell \(Z_0\) is the a.s. unique cell containing a fixed reference point, while the typical cell \(Z\) is defined via a rotation-invariant marking and Palm-distribution argument. Their expectations are linked by \[ \mathbb E[f(Z_0)] =\gamma_{\rm cell}\,\mathbb E\!\bigl[f(Z)\,\sigma_d(Z)\bigr] \] for rotation-invariant measurable \(f\) [1709.06522]. The spherical Urysohn inequality states that if \(K\in\mathcal K_g\) and \(C\subset S^d\) is a spherical cap with \(V_s(K)=V_s(C)\), then \[ U_1(K)\ge U_1(C), \] with equality if and only if \(K\) is a cap. A corresponding inequality holds for \(\Psi(K)\) 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 \(\pi/2-a_0\), then \[ U_1(K)\ge \bigl(1+\beta A_2(K)^2\bigr)U_1(C), \] where \(A_2(K)\) is the \(L^2\)-deviation of the radial function; and if \(r_s(K)\ge a\) and \(\Delta(K)\ge \varepsilon\), then \[ U_1(K)\ge \bigl(1+c(a,d)\varepsilon^2\bigr)U_1(B(e,a)) \] [1709.06522]. These deterministic inequalities feed directly into probabilistic deviation bounds. For suitable size functional \(\Sigma\) and deviation functional \(\theta\), there are constants \(c_1,c_2>0\) such that \[ \mathbb P\bigl(\theta(Z_0)\ge \varepsilon \mid \Sigma(Z_0)\ge a\bigr) \le c_1 \exp(-c_2\gamma), \] and the same form holds for the typical cell \(Z\). In the Voronoi case, \[ \mathbb P\bigl(\mathcal R(Z)-\rho(Z)\ge \varepsilon \mid \rho(Z)\ge a\bigr) \le C_{15}\exp(-C_{16}\gamma) \] [1709.06522]. The asymptotic regime is fundamentally different from the Euclidean one. For the Crofton cell and the typical cell, \[ \lim_{\gamma\to\infty}\frac1\gamma \ln \mathbb P\bigl(\Sigma(Z)\ge a\bigr) =-\omega_{d+1}T(a), \] where \[ T(a)=\min\{\Phi(K): \Sigma(K)\ge a\}. \] Because \(S^d\) is compact, there is no analogue of the Euclidean large-cell regime; the natural asymptotic is \(\gamma\to\infty\), and the resulting limit laws are exponential tails rather than Gaussian fluctuations [1709.06522]. 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 [2503.13455]. 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 \(u,v\sim{\rm Unif}[0,1]\), sets \[ \theta_s=\arccos(1-u(1-\cos\theta_{\max})),\qquad \phi_s=2\pi v \] for the spherical cap point, and \[ \rho_p=\sqrt{u}\,\rho_{\max},\qquad \phi_p=2\pi v \] for the planar point. If \[ \rho_{\max}=R_s\sin\theta_{\max},\qquad R_s\cos\theta_{\max}