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Given a spherical set of points, we consider the detection of cocircular subsets of the data. We distinguish great circles from small circles, and develop algorithms for detecting cocircularities of both types. The suggested approach is an extension of the Hough transform. We address the unique parameter-space quantization issues arising due to the spherical geometry, present quantization schemes, and evaluate the quantization-induced errors. We demonstrate the proposed algorithms by detecting cocircular cities and airports on Earth’s spherical surface. These results facilitate the detection of great and small circles in spherical images.

We propose novel parametric concentric multi-unimodal small-subsphere families of densities for p − 1 ≥ 2-dimensional spherical data. Their parameters describe a common axis for K small hypersubspheres, an array of K directional modes, one mode for each subsphere, and K pairs of concentrations parameters, each pair governing horizontal (within the subsphere) and vertical (orthogonal to the subsphere) concentrations. We introduce two kinds of distributions. In its one-subsphere version, the first kind coincides with a special case of the Fisher–Bingham distribution, and the second kind is a novel adaption that models independent horizontal and vertical variations. In its multisubsphere version, the second kind allows for a correlation of horizontal variation over different subspheres. In medical imaging, the situation of p − 1 = 2 occurs precisely in modeling the variation of a skeletally represented organ shape due to rotation, twisting, and bending. For both kinds,we provide new computationally feasible algorithms for simulation and estimation and propose several tests. To the best knowledge of the authors, our proposedmodels are the first to treat the variation of directional data along several concentric small hypersubspheres, concentrated near modes on each subsphere, let alone horizontal dependence. Using several simulations, we show that our methods are more powerful than a recent nonparametric method and ad hoc methods. Using data from medical imaging, we demonstrate the advantage of our method and infer on the dominating axis of rotation of the human knee joint at different walking phases.

This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets of the sphere, emphasizing both continuous and discrete settings. We first present a detailed geometric and analytical foundation for intrinsic quantization on the unit sphere, including definitions of great and small circles, spherical triangles, geodesic distance, Slerp interpolation, the Fréchet mean, spherical Voronoi regions, centroid conditions, and quantization dimensions. Building upon this framework, we develop explicit continuous and discrete quantization models on spherical curves, namely great circles, small circles, and great circular arcs—supported by rigorous derivations and pedagogical exposition. For uniform continuous distributions, we compute optimal sets of n-means and the associated quantization errors on these curves; for discrete distributions, we analyze antipodal, equatorial, tetrahedral, and finite uniform configurations, illustrating convergence to the continuous model. The central conclusion is that for a uniform probability distribution supported on a one-dimensional geodesic subset of total length L, the optimal n-means form a uniform partition and the quantization error satisfies Vn=L2/(12n2).The exposition emphasizes geometric intuition, detailed derivations, and clear step-by-step reasoning, making it accessible to beginning graduate students and researchers entering the study of quantization on manifolds. This article is intended as an expository and tutorial contribution, with the main emphasis on geometric reformulation and pedagogical clarity of intrinsic quantization on spherical curves, rather than on the development of new asymptotic quantization theory.

This article presents a comprehensive and analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric. Focusing on highly symmetric configurations on the unit sphere S2, we investigate three explicit models of discrete uniform distributions and derive closed-form expressions for their optimal quantizers and corresponding mean square quantization errors. (I) For N equally spaced points on the equator, we obtain exact error formulas for both divisible and non-divisible cases n∤N, demonstrating that optimal Voronoi cells form contiguous arcs with midpoint representatives. (II) For two antipodally symmetric small circles at latitudes ±ϕ0, each with M longitudes, we prove a no-cross-circle Voronoi phenomenon, establish symmetry-preserving optimality, and derive finite-sum error formulas together with sharp curvature-dependent bounds and asymptotics. (III) For a single small circle at latitude ϕ0, we obtain analogous exact error formulas and show that curvature reduces distortion by a factor of cos2ϕ0, while preserving the n−2 decay rate. Across all models, we rigorously establish the “block midpoint principle”: optimal Voronoi cells on a circle are contiguous azimuthal blocks, and their optimal representatives are the corresponding azimuthal midpoints. Numerical tables and illustrative figures highlight curvature effects and compare divisible and non-divisible cases. An algorithmic appendix provides pseudocode and a small, commented Python implementation to facilitate reproducibility. Written with didactic clarity while maintaining full mathematical rigor, this work bridges geometric intuition and analytic precision, providing explicit benchmark models that illuminate curvature effects and support further developments in quantization on curved manifolds.

Traditional celestial navigation adopts the intercept method or Sumner line method in order to solve the fix problems of celestial sight reduction. While an estimated position is often needed in order to complete celestial positioning, these methods are not rigorous and they have many shortcomings. Thus, this study has analyzed the geometry of the circle of equal altitude and the two-body problem. Furthermore, vector methods are proposed without using spherical trigonometry, in order to meet navigation needs. Finally, web mapping was used to display the results. The results show that the two methods that are proposed in this study had better accuracy, practical convenience, and faster computing speed than those which have been proposed by other studies. The two methods’ capability to avoid complicated manual calculations and chart work processes makes them highly suitable as an alternative to the use of global navigation satellite systems. These methods enable maritime school students to quickly understand the principles of celestial navigation through computing and graphical interfaces.

This article discusses a novel framework to analyze rotational deformations of real three-dimensional objects. The rotational deformations such as twisting or bending have been observed as the major variation in some medical applications, where the features of the deformed three-dimensional objects are directional data. We propose modeling and estimation of the global deformations in terms of generalized rotations of directions. The proposed method can be cast as a generalized small circle fitting on the unit sphere. We also discuss the estimation of descriptors for more complex deformations composed of two simple deformations. The proposed method can be used for a number of different three-dimensional object models. Two analyses of three-dimensional object data are presented in detail: one using skeletal representations in medical image analysis and the other from biomechanical gait analysis of the knee joint. Supplementary materials for this article are available online.

Sphericity is a measure of the closeness of shape of clastic pebbles, sand grains, etc. to that of a true sphere, as expressed by the cube root of the ratio of the volume of the particle to that of its circumscribing sphere, the Wadell sphericity index. It was introduced by the Swedish-American geologist, Hakon Wadell (1895–1962) (Wadell 1932). However, because of the difficulty of measuring the surface area of irregular solids, it was subsequently modified (Wadell 1935) to a projective measure: the ratio of the diameter of a circle whose area is equal to that of the grain to the diameter of the smallest circle circumscribing the grain which, in practice, is generally equal to the long-axis of the grain. The American mathematical geologist, William Christian Krumbein (1902–1979) (Krumbein 1941) introduced a method of estimating sphericity, based on approximating particle shape to that of a triaxial ellipsoid and using the ratios of the long (L), intermediate (I) and short (S) diameters, I/L and S/I. See also: roundness.

SUMMARY In the analysis of astrophysical data assumed to lie on the celestial sphere, new directional data techniques are necessary when the monitoring station is on the earth's surface. Weighted spherical random variables are introduced and investigated for the cases when the true underlying distributions are either uniform, or of the Fisher or Dimroth-Watson forms. The uniform distribution on a rotating cap is examined and numerical examples are given with data for the arrival directions of ultra-high energy cosmic rays.

A new distribution is proposed as a model for measurements of direction that cluster near a small circle, i.e. parallel of latitude, on the sphere. It is a generalization of both the Fisher-von Mises and Dimroth-Watson distributions. Methods of estimation and inference are discussed and an approximation that is valid for large values of the shape parameter is introduced. The use of the distribution is illustrated by the analysis of artificially generated data