Search Results (105)

This paper investigates the constrained particle dynamics on rotational surfaces within the 4-dimensional pseudo-Euclidean space $E_2^4$, characterized by its second-order metric signature of index 2. A comprehensive Lagrangian and Hamiltonian formulation is developed to construct the specific energy and specific angular momentum as conserved Noetherian charges along timelike geodesics. By integrating Clairaut’s theorem into the geodesic flow equations, explicit analytical expressions for these fundamental physical invariants are obtained. This work explores the structural relationship between the surface’s continuous rotational symmetries and the mechanical stability of the geodesic flow. A mathematical resolution for the signature transitions manifested via the appearance of the imaginary unit i on elliptic surfaces is provided through analytic continuation and distinct coordinate charts. Furthermore, by reducing the second-order geodesic flow to a one-dimensional energy balance equation, the exact effective potentials ($V_{eff}$) are derived, and the local orbital stability zones are analytically verified via second-order radial derivatives (${\partial }_s^2{V}_{eff}>0$). These embedded geometric configurations are shown to share qualitative features with the equatorial slices of rotating relativistic spacetimes. Consequently, they can serve as potential geometric toy-models for studying the dynamics of photon spheres, ergosphere oscillations, and innermost stable circular orbits in extreme gravitational fields. Full article

Curves constructed directly on point clouds are a core primitive in reverse engineering, product design, and point-based CAD; many workflows additionally require the curve to lie in a plane—e.g., as a section profile, inspection path, or design reference. This paper presents an algorithmic framework for computing free and plane-constrained geodesic curves directly on oriented point clouds, without any intermediate surface or mesh reconstruction. A geodesic-curvature-minimizing solver that combines a Newton/conjugate-gradient flow with directed projection, elliptic Gabriel neighborhoods, and Taubin smoothing forms the backbone; the plane-constrained problem is then reduced to a one-parameter pencil of planes through the endpoint chord and solved per plane by alternating projection onto the cloud and the plane, with a projection-only pre-lift and a penalized length objective that rejects sections floating off the cloud; the returned section is the best found over a sampled pencil of candidate planes. The returned sections are attached to the cloud within a small fraction of the mean sampling distance. All algorithms are given in pseudocode with convergence criteria and complexity estimates. Two parallel realizations of the plane search are developed and measured: a multithreaded CPU backend (about 3× over the serial scan) and a WebGPU backend that evaluates the whole plane pencil in a single compute dispatch. Accuracy is validated against the analytic conic sections of a cone and against cylinder and sphere benchmarks whose optimal plane is known in closed form; robustness is assessed under noise, non-uniform sampling, missing regions, outliers, and perturbed normals, and against both a slab-projection baseline and the conventional reconstruct-then-slice route. Five applications—shoe-last reverse engineering with a C² surface reconstruction, anthropometric girth measurement, medical transverse sectioning, dimensional metrology on industrial mold scans, and cleaning-path planning for a robotic surface-treatment task—demonstrate the plane-constrained geodesic curves in practice. Full article

This paper develops a unified Lorentzian framework for persistent rigid motions generated by slant helices in three-dimensional Minkowski space and investigates their geometric and kinematic properties. Persistence is characterized by the constancy of the pitch of the instantaneous twist associated with a one-parameter rigid motion in the Poincaré group $\mathrm{ISO}(2,1)$. Interpreting curves in the motion group as trajectories of rigid motions, we study Frenet–Serret and adapted frame motions determined by slant helices under different causal characters. Necessary and sufficient conditions are established for these frame motions to generate persistent Lorentzian motions. An explicit intrinsic relationship between the pitches of Frenet–Serret and adapted frame motions is obtained in terms of the geodesic curvature of the spherical image of the principal normal indicatrix, showing that persistence is governed by intrinsic curve invariants and is independent of the chosen moving frame. The geometric structure of persistent motions is further clarified through associated ruled surfaces. In particular, the pitch of a persistent motion is shown to coincide with the distribution parameter of the ruled surface associated with the corresponding frame motion. Illustrative examples are presented for different causal configurations. These results extend classical Euclidean theory of persistent rigid motions to the Lorentzian setting and provide a unified framework connecting curve theory, frame geometry, ruled surfaces, and Lorentzian kinematics. Full article

Inthis paper, we introduce a family of ruled surfaces called the osculating type (OT) ruled surfaces of a B-Lift curve. By taking into account an alternative frame on the surface, we obtain more simplified geometric invariants of the surface. We calculate the Gaussian and the mean curvatures of the OT ruled surface and determine the conditions in which this surface is minimal or flat. Moreover, we also determine the conditions necessary for any curve on an OT ruled surface to be an asymptotic curve, a curvature line, or a geodesic. We also present two numerical examples of the obtained results. Full article

Functional brain networks in schizophrenia (SZ) are often characterized by covariance-based measures, yet covariance matrices live on a curved geometric structure rather than in ordinary Euclidean space, complicating noise-robust inference from scalp EEG. We develop a Riemannian Geometry-based Adaptive Nonlinear Coupling Analysis (RGA-NCA) framework that integrates the affine-invariant Riemannian metric (AIRM), tangent space mapping (TSM), and an anatomically adaptive artifact rejection (AAAR) strategy accounting for regional signal-to-noise heterogeneity. The framework is grounded in the observation that Euclidean summaries of symmetric positive definite matrices are sensitive to noise-driven volume inflation, whereas geodesic distances on the manifold emphasize shape deformation. RGA-NCA was evaluated on four benchmark dynamical systems, a supplementary multichannel EEG-like sample covariance simulation, and a public button-tone SZ/HC EEG dataset associated with the auditory feedback paradigm described by Ford et al. (81 subjects; 49 SZ, 32 healthy controls). Compared with Euclidean and linear baselines, RGA-NCA showed lower sensitivity to noise-driven distance distortion and yielded clearer group-level contrasts in the tested ROI analyses; all four pre-specified frontotemporal and parietal channel pairs remained significant after Benjamini–Hochberg FDR correction. The resulting patterns are consistent with reduced long-range connectivity together with localized hyper-synchronization-like effects in SZ. Quantitatively, the Riemannian structural sensitivity index $(\mathrm{sim}=\exp (-d^2/4))$ remained high across all tested SNR levels (−20 to +10 dB; 50 Monte Carlo trials per level; range 0.936–0.964), with only a 0.026 endpoint change between +10 and −20 dB, whereas the Euclidean metric fell from 0.922 at +10 dB to 0.000 at −20 dB. These findings support Riemannian modeling as a candidate strategy for noisy covariance-based neural data, pending validation in larger independent cohorts. Full article

In this paper, we study magnetic curves associated with a contact magnetic field on almost cosymplectic manifolds. In particular, we consider locally conformal almost cosymplectic manifolds and almost $\alpha$-cosymplectic f-manifolds. The magnetic field is defined by the fundamental 2-form $\Omega$, with the corresponding Lorentz force determined by the structure tensor $\phi$. We classify normal magnetic curves in these settings and show that they are either geodesics, Legendre $\phi$-circles, or $\phi$-helices of order three. We further investigate slant curves and derive conditions relating their geometry to the structure functions of the manifold. These results generalize known classifications of magnetic curves in contact and cosymplectic geometries and provide a unified treatment for the considered classes of manifolds. Full article

In this study, we examine a length preserving geodesic curvature difference flow for smooth strictly horocyclically convex simple closed curves in the hyperbolic plane ${\mathbb{H}}^2$. Given an initial curve ${\gamma }_1$ and a target curve ${\gamma }_2$ of the same hyperbolic length, we evolve ${\gamma }_1$ by a normal speed given by the difference of the reciprocals of geodesic curvatures evaluated at points with the same outward unit normal, together with a time-dependent scalar term $\mathrm{\Gamma }\left(t\right)$ chosen to preserve the hyperbolic length. Using Leichtweiβ’s hyperbolic support function and Howe’s curvature formula, the flow is reformulated as a quasilinear uniformly parabolic equation on ${\mathbb{S}}^1$ with a nonlocal term $\mathrm{\Gamma }\left(t\right)$. We prove short-time existence, uniqueness, and preservation of strict horocyclic convexity. Linearizing the support function equation at the target support function yields a uniformly elliptic operator whose kernel contains the infinitesimal isometry directions. Under a spectral gap assumption on a normalized slice transverse to the isometry orbit, we prove global existence and exponential convergence for initial data sufficiently close to the target curve. In the last section, this assumption is verified explicitly when the target curve is a geodesic circle. Full article

We present a geodesic dynamics framework for discrete-time state evolution on the unit sphere $S^{N-1}$ that maintains exact unit-norm constraints through Riemannian exponential mapping. Given an input sequence and an initial state, the method constructs trajectories by projecting inputs to tangent spaces and updating states along geodesics, incorporating temporal memory via approximate parallel transport of velocity directions. Unlike traditional approaches requiring post hoc normalization of linear updates, the geodesic formulation preserves $\parallel {\mathbf{x}}_t\parallel =1$ to machine precision while eliminating explicit $N\times N$ transition matrices in favor of $D\times N$ input embeddings when the intrinsic input dimension D is much smaller than the ambient dimension N. The update corresponds to a first-order exponential integrator on the sphere. We establish local Lipschitz continuity of the exponential map on positively curved manifolds with careful treatment of basepoint dependence, derive perturbation bounds showing linear-to-exponential growth transitions via Grönwall-type estimates, and we prove third-order asymptotic equivalence with normalized linear systems under appropriate scaling. Numerical experiments on synthetic data validate exact norm preservation over extended time horizons, confirm theoretical perturbation growth predictions, and demonstrate the effectiveness of the temporal memory mechanism in reducing long-horizon prediction errors. The framework provides a principled geometric approach for applications requiring exact directional or compositional constraints. Full article

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 ${\mathbb{S}}^2$, 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 $n\nmid N$, demonstrating that optimal Voronoi cells form contiguous arcs with midpoint representatives. (II) For two antipodally symmetric small circles at latitudes $\pm {\varphi }_0$, 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 ${\varphi }_0$, we obtain analogous exact error formulas and show that curvature reduces distortion by a factor of ${cos}^2{\varphi }_0$, 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. Full article

This article provides a geometric characterization of the geodesic boundary for surfaces invariant under parabolic isometries in ${\mathbb{H}}^2\times \mathbb{R}$. We present an alternative, constructive proof for the existence of minimal surfaces with rectangular asymptotic boundaries by utilizing a specific family of invariant surfaces. Furthermore, we generalize these existence results to surfaces with constant mean curvature $H\in (0,1/2)$. By analyzing the variation in the relative asymptotic height, we establish the existence of properly embedded H-surfaces whose geodesic boundary is a rectangle of arbitrary height, provided it exceeds a specific lower bound determined by the parabolic solution. Full article

Complex systems—ranging from biological organisms to turbulent fluids—exhibit multiscale heterogeneity and intermittency that traditional, differentiable calculus fails to adequately capture. Therefore, we propose a mathematical framework for analyzing complex system dynamics by assimilating the trajectories of structural units to continuous but non-differentiable multifractal curves. Utilizing the scale covariance principle, the authors recast the conservation of momentum as a geodesic equation within a multifractal space. This approach naturally separates the complex velocity field into differentiable and non-differentiable scale resolutions, where the balance of multifractal acceleration, convection, and dissipation is parametrized by a singularity spectrum f(α). We also discuss broad interdisciplinary implications, because, in our opinion, non-differentiability can enhance predictive capabilities in various fields such as oncology, pharmacology, and geophysics. Full article

Bertrand curve pairs share the same principal normals, creating a geometric symmetry useful in design and modeling. The geometry of a surface can be characterized and studied through three types of characteristic curves: geodesics, curvature lines, and asymptotic curves. We introduce a method to construct corresponding surface family pairs from a Bertrand curve pair, ensuring that both curves serve as the same type of characteristic curve on each surface family, thereby extending curve symmetry to surface symmetry. We build surface pairs by linearly combining the Frenet frame of a Bertrand curve, with coefficients acting as shape functions. We establish necessary and sufficient conditions that these functions must satisfy to guarantee that the Bertrand curves become the same characteristic type on both surface families. This provides flexible control over surface geometry and curve type. We further derive conditions for developable surface pairs, proving that no developable pair can contain a twisted Bertrand curve as a curvature line or asymptotic curve. To illustrate this, we construct surface pairs from a circular helix, and the resulting surfaces exhibit an aesthetically pleasing symmetry, demonstrating the flexibility and interactivity of our framework. Full article

In this research, we establish a precise correspondence between the theory of logarithmic connections on principal G-bundles over compact Riemann surfaces and the geometric formulation of control systems on curved manifolds, providing a novel differential–geometric framework for analyzing optimal control problems with non-holonomic constraints. By characterizing control systems through the geometric structure of flat connections with logarithmic singularities at marked points, we demonstrate that optimal trajectories correspond precisely to horizontal lifts with respect to the connection. These horizontal lifts project onto geodesics on the punctured surface, which is equipped with a Riemannian metric uniquely determined by the monodromy representation around the singularities. The main geometric result proves that the isomonodromic deformation condition translates into a compatibility condition for the control system. This condition preserves the conjugacy classes of monodromy transformations under variations of the marked points, and ensures the existence and uniqueness of optimal trajectories satisfying prescribed boundary conditions. Furthermore, we analyze systems with non-holonomic constraints by relating the constraint distribution to the kernel of the connection form, showing how the degree of non-holonomy can be measured through the failure of integrability of the associated horizontal distribution on the principal bundle. As an application, we provide computational implementations for $\mathrm{SL}(2,\mathbb{C})$ connections over hyperbolic Riemann surfaces with genus $g\ge 2$, explicitly constructing the monodromy-induced metric via the Poincaré uniformization theorem and deriving closed-form expressions for optimal control strategies that exhibit robust performance characteristics under perturbations of initial conditions and system parameters. Full article

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 $V_n=L^2/\left(12n^2\right)$.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. Full article

Diffusion on curved surfaces deviates from the flat case due to geometrical corrections in the evolution of its moments, such as the geodesic mean square displacement. Moreover, anomalous diffusion is widely used to model transport in disordered, confined, or crowded environments and can be described by a temporal subordination scheme, leading to a time-fractional diffusion equation. In this work, we analyze the dynamics of time subordinated anomalous diffusion on curved surfaces. By using a generalized Taylor expansion with fractional derivatives in the Caputo sense, we express the moments as a temporal power series and show that the anomalous exponent couples with curvature terms, leading to a competition between geometric and anomalous effects. This coupling indicates a mechanism through which curvature modulates anomalous transport. Full article