Search Results (74)

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

The rapid expansion of telecommunication infrastructure in developing countries has increased the demand for sustainable strategies to deploy field engineers in tower maintenance operations. Traditional approaches often neglect spatial factors, resulting in inefficient workforce allocation, excessive travel, and higher carbon emissions. This study develops an applied geospatial deployment framework that integrates spatial analysis with sustainable supply chain management (SSCM) principles to support operational decision-making in resource-constrained telecommunication maintenance environments. Using publicly available tools, tower and homebase coordinates were mapped and analyzed through Haversine-based geodesic distance calculations, with a comparative assessment against Euclidean approximation, while incorporating operational constraints such as service time per tower, available personnel, and work-hour limitations. The results indicate that the existing two-homebase deployment strategy leads to unbalanced workloads and unnecessary travel distances. By introducing a cluster-based restructuring using k-means to identify four sub-homebases, the proposed approach reduces total round-trip travel distance from 9120 km to 5913 km per maintenance cycle, representing a 35.2% reduction. This distance reduction corresponds to an estimated saving of approximately 593 kg of CO₂ emissions per maintenance cycle, representing an operational-scale reduction in travel-related emissions based on distance-derived fuel consumption modeling and assuming typical fuel efficiency for service vehicles. In addition, the optimized spatial configuration enables a more equitable distribution of engineers and reduces travel-related fatigue. These findings demonstrate the value of integrating geospatial optimization with sustainable supply chain management by aligning operational efficiency with quantifiable environmental and social sustainability outcomes. The proposed framework offers a replicable, low-cost, and data-driven solution for telecommunication infrastructure providers seeking to enhance the sustainability of field service operations in resource-constrained environments. Full article

Visual object tracking from unmanned aerial vehicles (UAVs) demands both high accuracy and computational efficiency for real-time deployment on resource-constrained platforms. While state space models (SSMs) offer linear computational complexity, existing methods face critical deployment challenges. They rely on the HiPPO framework with complex discretization procedures and employ hardware-aware algorithms optimized for high-performance GPUs, which introduce deployment overhead and are difficult to transfer to edge platforms. Additionally, their fixed polynomial bases may cause information loss for tracking features with complex geometric structures. We propose LF-SSM, a lightweight HiPPO (High-order Polynomial Projection Operators)-free state space model that reformulates state evolution on Riemannian manifolds. The core contribution is the Geodesic State Module (GSM), which performs state updates through tangent space projection and exponential mapping on the unit sphere. This design eliminates complex discretization and specialized hardware kernels while providing adaptive local coordinate systems. Extensive experiments on UAV benchmarks demonstrate that LF-SSM achieves state-of-the-art performance while running at 69 frames per second (FPS) with only 18.5 M parameters, demonstrating superior efficiency for real-time edge deployment. Full article

We introduce a geometric framework in which classical transforms are represented as coordinate charts on a symbolic manifold. The construction defines symbolic curvature (κ), strain (τ), compressibility (σ), and the ratio Γ = κ/τ, which together provide a diagnostic coordinate system for comparing representational stability across chart transitions. Within this setting, transforms such as Fourier, Laplace, wavelet, Jordan, and polynomial projection can be treated as charts connected by transition maps that preserve Γ on specified domains. We also introduce a symmetric positive-definite metric tensor $G_{ab}$ to quantify displacement in the invariant coordinates and to formalize minimal-effort paths (geodesics) under modeling assumptions stated in the text. The resulting framework provides a reproducible screening method for evaluating transform stability, diagnosing closure failure, and comparing transform behavior under a shared set of invariants. Full article

We introduce and study a new class of noncyclic Chatterjea-type C-nonexpansive mappings in geodesic spaces. We establish a notable existence theorem for best proximity pairs by employing the pivotal geometric property of proximal normal structure within the framework of reflexive Busemann convex spaces. Moreover, we investigate minimal invariant sets associated with these mappings and derive a generalization of the Goebel–Karlovitz lemma. Our main contribution extends this fundamental result to geodesic spaces with property UC, thereby providing a significant generalization of the classical theorem for the case of Chatterjea-type C-nonexpansive mappings. Full article

This paper presents a comparative study on the suitability of free-boundary surface parameterization techniques for generating trajectories on 3D surfaces. The approach maps a 3D surface to a 2D parametric domain through four parameterization methods: Least-Squares Conformal Mapping, Boundary First Flattening, As-Rigid-As-Possible, and Conformal Equivalence of Triangular Meshes. Structured trajectory patterns are generated in the 2D domain and projected back to 3D. We introduce center-to-boundary geodesic deviation measure, which yields a deviation profile over the boundary loop and reflects how well central alignment is preserved under each parameterization method. The results highlight differences in distortion and geodesic preservation, reflecting the suitability of methods for path generation. Full article

Embedding text on 3D triangular meshes is essential for conveying semantic information and supporting reliable identification and authentication. However, existing methods often fail to incorporate the geometric properties of the underlying mesh, resulting in shape inconsistencies and visual artifacts, particularly in regions with high curvature. To overcome these limitations, we present GeoText, a framework for generating 3D text directly on triangular meshes while faithfully preserving local surface geometry. In our approach, the control points of TrueType Font outlines are mapped onto the mesh along a user-specified placement curve and reconstructed using geodesic Bézier curves. We introduce two mapping strategies—one based on a local tangent frame and another based on straightest geodesics—that ensure natural alignment of font control points. The reconstructed outlines enable the generation of embossed, engraved, or independent 3D text meshes. Unlike Boolean-based methods, which combine text meshes through union or difference and therefore fail to lie exactly on the surface—breaking the symmetry between embossing and engraving—our offset-based approach ensures a symmetric relation: positive offsets yield embossing, whereas negative offsets produce engraving. Furthermore, our method achieves robust text generation without self-intersections or inter-character collisions. These capabilities make GeoTextwell suited for applications such as 3D watermarking, visual authentication, and digital content creation. Full article

Background/Objectives: Synaptic plasticity is fundamental to learning and memory, yet classical models such as Hebbian learning and spike-timing-dependent plasticity often overlook the distributed and wave-like nature of neural activity. We present a computational framework grounded in Scale Relativity Theory (SRT), which describes neural propagation along fractal geodesics in a non-differentiable space-time. The objective is to link nonlinear wave dynamics with the emergence of structured memory representations in a biologically plausible manner. Methods: Neural activity was modeled using nonlinear Schrödinger-type equations derived from SRT, yielding complex wave solutions. Synaptic plasticity was coupled through a reaction–diffusion rule driven by local activity intensity. Simulations were performed in one- and two-dimensional domains using finite difference schemes. Analyses included spectral entropy, cross-correlation, and Fourier methods to evaluate the organization and complexity of the resulting synaptic fields. Results: The model reproduced core neurobiological features: localized potentiation resembling CA1 place fields, periodic plasticity akin to entorhinal grid cells, and modular tiling patterns consistent with V1 orientation maps. Interacting waveforms generated interference-dependent plasticity, modeling memory competition and contextual modulation. The system displayed robustness to noise, gradual potentiation with saturation, and hysteresis under reversal, reflecting empirical learning and reconsolidation dynamics. Cross-frequency coupling of theta and gamma inputs further enriched trace complexity, yielding multi-scale memory structures. Conclusions: Wave-driven dynamics in fractal space-time provide a hypothesis-generating framework for distributed memory formation. The current approach is theoretical and simulation-based, relying on a simplified plasticity rule that omits neuromodulatory and glial influences. While encouraging in its ability to reproduce biological motifs, the framework remains preliminary; future work must benchmark against established models such as STDP and attractor networks and propose empirical tests to validate or falsify its predictions. Full article

Conformal mappings between Riemannian spaces ${\overline{\mathbb{R}}}_N$ and ${\mathbb{R}}_N$ are defined by the explicit transformation of the metric tensor of the space ${\overline{\mathbb{R}}}_N$ to the metric tensor of the space ${\mathbb{R}}_N$. Geodesic mapping between these two Riemannian spaces is a transformation that transforms any geodesic line of the space ${\overline{\mathbb{R}}}_N$ to a geodesic line of the space ${\mathbb{R}}_N$. In this research, we defined an m-conformal line of a Riemannian space, which is geodesic if $m=0$. Based on this definition, we involved the concept of $(\overline{m},m)$-conformal mapping as a transformation ${\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ in which any $\overline{m}$-conformal line of the space ${\overline{\mathbb{R}}}_N$ transforms to an m-conformal line of the space ${\mathbb{R}}_N$. The result of this research is the establishment of three invariants for these mappings. At the end of this research, we gave an example of a scalar geometrical object which may be used in physics. Full article

Non-solid infill generation in Non-Planar Additive Manufacturing (NPAM) is still an open problem. This is due to mathematical complexities from curvature distortion, as well as bridging limitations inherent in some NPAM processes. Providing solutions to this problem may result in significant energy, build cycle time, and cost savings. In this context, the goal of this paper is to define a workflow for the generation of non-solid infill paths with quasi-uniform density within the layer. This was performed by defining the build geometry through an axisymmetric embedded map methodology, and the infill points were distributed via a geodesic repulsion energy-based algorithm. In addition to these core algorithms, several numeric optimizations were implemented to reduce runtime. The algorithm has been tested on several build platform geometries and slice polygons. The results were satisfactory, achieving a homogeneous kernel density distribution for all cases and reductions in geodesic distance standard deviations of around 70%. A first iteration of a path planning algorithm was also implemented to showcase the intended final results. This methodology is to be combined with other Design for Non-Planar Additive Manufacturing techniques to enable applications in the biomedical field, automotive and aerospace industry, or rapid mold manufacturing. Full article

The convex optimization problems have been considered by many researchers on geodesic spaces. In these problems, the resolvent operators play an important role. In this paper, we propose new resolvents on geodesic spaces, and we show that they have better properties than other known resolvent operators. Full article

In the paper, we consider holomorphically projective mappings of n-dimensional pseudo-Riemannian Kähler and hyperbolic Kähler spaces. We refined the fundamental linear equations of the above problems for metrics of differentiability class $C^2$. We have found the conditions for n complete geodesics and their image that must be satisfied for the holomorphically projective mappings to be trivial, i.e., these spaces are rigid with precision to affine mappings. Full article

Conformal maps preserve angles and maintain the local shape of geometric structures. The osculating curve plays an important role in analyzing the variations in curvature, providing a detailed understanding of the local geometric properties and the impact of conformal transformations on curves and surfaces. In this paper, we study osculating curves on regular surfaces under conformal transformations. We obtained the conditions required for osculating curves on regular surfaces R and $\tilde{R}$ to remain invariant when subjected to a conformal transformation $\psi :R\to \tilde{R}$. The results presented in this paper reveal the specific conditions under which the transformed curve $\tilde{\sigma }=\psi \circ \sigma$ preserves its osculating properties, depending on whether $\tilde{\sigma }$ is a geodesic, asymptotic, or neither. Furthermore, we analyze these conditions separately for cases with zero and non-zero normal curvatures. We also explore the behavior of these curves along the tangent vector $T_{\sigma }$ and the unit normal vector $P_{\sigma }$. Full article

With the advancement of service robot technology, the demand for higher boundary precision in indoor semantic segmentation has increased. Traditional methods of extracting Euclidean features using point cloud and voxel data often neglect geodesic information, reducing boundary accuracy for adjacent objects and consuming significant computational resources. This study proposes a novel network, the Euclidean–geodesic network (EGNet), which uses point cloud–voxel–mesh data to characterize detail, contour, and geodesic features, respectively. The EGNet performs feature fusion through Euclidean and geodesic branches. In the Euclidean branch, the features extracted from point cloud data compensate for the detail features lost by voxel data. In the geodesic branch, geodesic features from mesh data are extracted using inter-domain fusion and aggregation modules. These geodesic features are then combined with contextual features from the Euclidean branch, and the simplified trajectory map of the grid is used for up-sampling to produce the final semantic segmentation results. The Scannet and Matterport datasets were used to demonstrate the effectiveness of the EGNet through visual comparisons with other models. The results demonstrate the effectiveness of integrating Euclidean and geodesic features for improved semantic segmentation. This approach can inspire further research combining these feature types for enhanced segmentation accuracy. Full article

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely $\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$, represented by the Riemann surfaces which intersect along the curve $\left(M_1,g_1\right){\displaystyle \cap }\left(M_2,g_2\right)=ℒ$ were addressed. Curve $ℒ$ does not contain geodesic lines in either of the manifolds$\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$. Consider six points located on the $ℒ:$ $\left\{1,\dots 6\right\}\subset ℒ$. The points $\left\{1,\dots 6\right\}\subset ℒ$ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold $\left(M_1,g_1\right)$/red links, and, alternatively, with the geodesic lines belonging to the manifold $\left(M_2,g_2\right)$/green links. Points $\left\{1,\dots 6\right\}\subset ℒ$ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented. Full article