Computational Geometry Research Articles

Abstract The theory of forbidden 0–1 matrices generalizes Turán-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parameter twin width . The foremost open problem in this area is to resolve the Pach-Tardos conjecture from 2005, which states that if a forbidden pattern $$P\in \{0,1\}^{k\times l}$$ P ∈ { 0 , 1 } k × l is acyclic , meaning it is the bipartite incidence matrix of a forest, then $$\operatorname {Ex}(P,n) = O(n\log ^{C_P} n)$$ Ex ( P , n ) = O ( n log C P n ) , where $$\operatorname {Ex}(P,n)$$ Ex ( P , n ) is the maximum number of 1s in a P -free $$n\times n$$ n × n 0–1 matrix and $$C_P$$ C P is a constant depending only on P . This conjecture has been confirmed on many small patterns, specifically all P with weight at most 5, and all but two with weight 6. The main result of this paper is a clean refutation of the Pach-Tardos conjecture. Specifically, we prove that $$\operatorname {Ex}(S_0,n),\operatorname {Ex}(S_1,n) \ge n2^{\Omega (\sqrt{\log n})}$$ Ex ( S 0 , n ) , Ex ( S 1 , n ) ≥ n 2 Ω ( log n ) , where $$S_0,S_1$$ S 0 , S 1 are the outstanding weight-6 patterns. We also prove sharp bounds on the entire class of alternating patterns $$(P_t)$$ ( P t ) , specifically that for every $$t\ge 2$$ t ≥ 2 , $$\operatorname {Ex}(P_t,n)=\Theta (n(\log n/\log \log n)^t)$$ Ex ( P t , n ) = Θ ( n ( log n / log log n ) t ) . This is the first proof of an asymptotically sharp bound that is $$\omega (n\log n)$$ ω ( n log n ) .

The reliable characterization of fabric anisotropy in concrete aggregates is critical for understanding the mechanical behavior and durability of concrete. The accurate segmentation of aggregates is essential for anisotropy assessment. However, conventional threshold-based segmentation methods exhibit high sensitivity to noise, while deep learning approaches are often constrained by the scarcity of annotated data. To address these challenges, this study introduces the Segment Anything Model (SAM) for automated aggregate segmentation, leveraging its remarkable zero-shot generalization capabilities. In addition, a novel quantification technique integrating computational geometry with second-order Fourier series is proposed to evaluate both the magnitude and orientation of fabric anisotropy. Extensive experiments conducted on a self-constructed concrete aggregate dataset demonstrated the effectiveness and accuracy of the proposed method. The process incorporates domain-specific image preprocessing using Contrast Limited Adaptive Histogram Equalization (CLAHE) to enhance the input quality for the SAM. The SAM achieves an F1-score of 0.842 and an intersection over union (IoU) of 0.739, with mean absolute errors of 4.15° for the orientation and 0.025 for the fabric anisotropy. Notably, optimal segmentation performance is observed when the SAM’s grid point parameter is set to 32. These results validate the proposed method as a robust, accurate, and automated solution for quantifying concrete aggregate anisotropy, providing a powerful tool for microstructure analysis and performance prediction.

Modeling the partially detected backside reflectance of transparent substrates in reflectance microspectroscopy.

Recent advancements in deep learning-based geophysical inversion have drawn considerable attention. Most of these inversions are supervised, which requires the creation of training models that capture as much prior geological information as is available in an area of interest. However, creating such geologically informed training models is challenging because some geological knowledge is difficult to be expressed in mathematical terms. Moreover, geological prior information is not always tied to specific spatial locations. To address these challenges, we propose a novel method based on alpha shapes, a concept from computational geometry, to generate training models that can easily integrate five key types of geological prior information, namely, (1) top boundaries, (2) dip angles, (3) surface outcrop contacts, (4) mineralization zones intersected by drillholes, and (5) measured physical property values on rock samples. We present three distinct scenarios to demonstrate how the proposed method can be employed to systematically generate geologically informed training models. We also show that deep generative models, such as the conditional variational autoencoder, trained on these geologically informed models, can not only output inversion results that align with prior geological knowledge but also quantify the associated uncertainties. To validate our approach, we apply it to a set of magnetic measurements collected in Qinghai Province, China, for the exploration of Cu-Mo critical mineral deposits. The resulting susceptibility models reveal a major dipping structure that is consistent with both surface geology and the magnetic data. The proposed method offers a flexible and unified framework for generating geologically informed training models for deep learning-based geophysical inversions.

We study linear relations between color-ordered all-plus amplitudes at one loop in Yang-Mills theory. We show that on general grounds, there are ( n − 1 ) ! / 2 − 2 relations for n ≥ 5 , leaving only two independent color-ordered amplitudes. We present two complementary approaches to finding such relations; one using numerical linear algebra and the other using syzygies in computational algebraic geometry. We obtain explicit forms for all relations through n = 7 . We also study relations for the tree-level maximally helicity violating amplitudes through n = 8 . The latter relations include the well-known color and Bern-Carrasco-Johansson identities.

We report the first implementation of spin-flip time-dependentdensity functional theory (SF-TDDFT) within the Tamm-Dancoff approximationin the Sternheimer formulation including the use of the noncollinearkernel. The noncollinear kernel was stabilized by introducing a screeningmethod for the numerical integration, realizing a robust scheme ofexcited energy and gradient calculations of SF-TDDFT using generalizedgradient approximation functionals. The implementation is evaluatedby benchmark calculations of vertical excitation energies and optimizedmolecular geometries. The benchmark for vertical excitations consistsof 19 excitations with high level of theory reference data from theQUESTDB. An underestimation of vertical excitation energies was observedfor the PBE and PBE0 functionals, as seen by their average deviationsof −0.3 eV. The benchmark for optimized geometries consistsof 25 optimized structures with high level of theory, comprising CCSD,CISD, and FCI data, and 10 reference structures optimized with otherimplementations of collinear and noncollinear SF-TDDFT. The optimizedstructures using PBE and PBE0, with a noncollinear kernel, were foundto be close to the high-level reference structures, with mean deviationsof 0.010 and −0.004 Å, respectively. The extension tothe auxiliary density matrix method (ADMM) is also presented. We foundan average deviation of 0.003 Å in the calculated bond lengthswhen employing the ADMM for the PBE0 functional.

Computational Geometry Based on Quantum Secure Multi-Party Summation and Multiplication

A geometric computation of cohomotopy groups in codegree one

Monte Carlo (MC) neutron transport provides detailed estimates of radiological quantities within fission reactors. This involves tracking individual neutrons through a computational geometry. CPU-based MC codes use multiple polymorphic tracker types with different tracking algorithms to exploit the repeated configurations of reactors, but virtual function calls have high overhead on the GPU. The Shift MC code was modified to support GPU-based tracking with three strategies: dynamic polymorphism with virtual functions, static polymorphism, and a single tracker type with tree-based acceleration. On the Frontier supercomputer these methods achieve 77.8%, 91.2%, and 83.4%, respectively, of the tracking rate obtained using a specialized tracker optimized for rectilinear-grid-based reactors. This indicates that all three methods are suitable for typical reactor problems in which tracking does not dominate runtime. The flexibility of the single tracker method is highlighted with a hexagonal-grid microreactor problem, performed without hexagonal-grid-specific tracking routines, providing a 2.19× speedup over CPU execution.

Abstract. Beside smart Information Management, Building Information Modeling (BIM) focuses on three-dimensional planning and model coordination. However, 3D CAD/BIM software does not conceptualize geodetic coordinates, which causes systematic deviations between geospatial surveys and 3D BIM Model. In local projects, the use of Cartesian coordinates (not curved geodetic coordinates!) is indispensable, due to the high precision demands in computational geometry. The solution is to convert geodetic coordinates into an optimized coordinate reference system. This procedure has been standardized and implemented for 6,326 traffic stations of the German Railway, minimizing systematic deviations and using the DB REF geodetic datum for consistent referencing the alignment of the railway tracks. The developed approach enables the precise use of geometric models (CAD/BIM, high quality 3D point clouds, measured surveys) less than 5ppm, facilitates conversion to other coordinate reference systems using GIS-standard tools, and allows construction projects to be directly staked out from the BIM-model.

Debates on the subjective criteria for evaluating Surgical Milestones, such as achieving the critical view of safety (CVS) during cholecystectomy, remain a prominent focus and challenge in the field of surgical data science. In this study, we computed anatomical metrics with machine learning tools and investigated the relationship between these objective anatomical metrics and subjective criteria for CVS achievement. We implemented and calibrated a zero-shot monocular depth estimation model for endoscopic images from cholecystectomies. These depth measures were integrated with human-annotated segmentation masks of three key anatomical structures relevant to CVS: cystic duct, cystic artery, and gallbladder. Computational geometry techniques were then employed to extract structure-specific depth distributions and compute two anatomical metrics: diagonal length and surface area. We tested for significant differences in these case-wise metrics, grouped by human-annotated CVS status. 2256 frames (35 cases) were graded on CVS criteria, of which 343 frames (17 cases) met all three CVS criteria and 384 frames (17 cases) met no CVS criteria. The calibrated depth model achieves 0.063 on absolute relative error and 0.774 on squared relative error in the metric measurement. The diagonal length and surface area of both the cystic duct and cystic artery were significantly larger when all CVS criteria were met. On average, the cystic duct (cystic artery) length was 7.6mm (13.6mm) longer when CVS criteria was met. In this study, we presented a pipeline that generates anatomical measures from monocular endoscopic images utilizing a depth estimation model, anatomical segmentation masks, and computational geometry. The diagonal length and surface area of the cystic duct and cystic artery were found to be significantly larger in cases where all CVS criteria are met; this lends support to the use of these anatomical metrics as objective grounds for assessing CVS achievement.

We address the question of numerically simulating the coupling of diffusion, advection and one-speed linear transport, with a specific focus on managing geometrical complexity. We base our work on recent advances from the computer graphics community, which has developed Monte Carlo algorithms simulating linear radiation transport in physically realistic scenes, with numerical costs that remain unaffected by geometrical refinement: adding more details to the scene description does not impact the computation time. The resulting benefits in terms of engineering flexibility are already fully integrated into the cinema industry and are gradually being adopted by the video game industry. Here we demonstrate that the same insensitivity to the geometric complexity can be achieved when considering not only one-speed linear transport, but also its coupling with diffusion and advection. In this case, pure linear-transport paths are replaced with advection-diffusion/linear-transport paths, which are composed of subpaths. Each subpath represents one of the three physical phenomena, and coupling is handled by switching from one subpath (i.e. phenomenon) to another. This approach is illustrated using a porous medium involving up to 10,000 pores, with the computation time being strictly independent of the number of pores, showing its ability to facilitate engineering calculations in complex geometries.

Voronoi partitioning is a fundamental geometric concept with applications across computational geometry, robotics, optimization, and resource allocation. While Euclidean distance is the most commonly used metric, alternative distance functions can significantly influence the shape and properties of Voronoi cells. This paper presents a comprehensive mathematical analysis of various distance metrics used in Voronoi partitioning, including Euclidean, Manhattan, Minkowski, weighted, anisotropic, and geodesic metrics. We analyze their mathematical formulations, geometric properties, topological implications, and computational complexity. This work aims to provide a theoretical framework for selecting appropriate metrics for Voronoi-based modeling in diverse applications.

ABSTRACTIntroductionGiven a text, rank and select queries return the number of occurrences of a character up to a position (rank) or the position of a character with a given rank (select). These queries have applications in compression, computational geometry, and most notably pattern matching in the form of the backward search, which is the backbone of many compressed full‐text indices. Currently, in practice, for text over non‐binary alphabets, the wavelet tree is probably the most used data structure for rank and select queries.ObjectiveThe goal of this work is to design techniques that accelerate rank, select, and access queries while retaining the space efficiency of both uncompressed and compressed representations.MethodsTo this end, we change the underlying tree structure from a binary tree to a quaternary tree and reduce cache misses by approximating rank queries using a predictive model to prefetch all data required for the actual rank query. Finally, we also extend our approach to Huffman‐shaped wavelet trees.ResultOur methods allowed us to achieve speedups of up to a factor of two for access and select queries and up to three for rank queries compared to the SDSL implementation. With Huffman‐shaped wavelet trees, we achieved up to three times faster access and select, and 3.8 times faster rank. In addition, our approach still achieves a reduction of space usage by up to 30% for compressible datasets.ConclusionBy changing the tree structure and using predictive prefetching, we obtain a data structure that substantially outperforms other standard wavelet tree implementations, delivering significantly faster query performance while also occupying compact space.

Achieving high-fidelity and robust qubit manipulations is a crucial requirement for realizing fault-tolerant quantum computation. Here, we demonstrate a single-hole spin qubit in a germanium quantum dot and characterize its control fidelity using gate set tomography. The maximum control fidelities reach 97.48%, 99.81%, 99.88% for the I, X/2 and Y/2 gate, respectively. These results reveal that off-resonance noise during consecutive I gates in gate set tomography sequences severely limits qubit performance. Therefore, we introduce geometric quantum computation to realize noise-resilient qubit manipulation. The geometric gate control fidelities remain above 99% across a wide range of Rabi frequencies. The maximum fidelity surpasses 99.9%. Furthermore, the fidelities of geometric X/2 and Y/2 (I) gates exceed 99% even when detuning the microwave frequency by ± 2.5 MHz (± 1.2 MHz), highlighting the noise-resilient feature. These results demonstrate that geometric quantum computation is a potential method for achieving high-fidelity qubit manipulation reproducibly in semiconductor quantum computation.

In the context of the deep integration of digital art and geometric computing, this paper proposes a digital art pattern generation method with arbitrary quadrilateral tiling. The aim is to break through the limitations of traditional fixed tiling templates in terms of adaptability to irregular tiling shapes, controllability of local deformations, and naturalness of boundary transitions. By decoupling the topological stability of quadrilaterals from deformation parameters and combining the Coons surface interpolation method, a smooth invariant mapping for the fundamental region of arbitrary quadrilaterals is constructed, solving the seamless splicing problem of irregular fundamental region. This method supports real-time editing of quadrilateral shape and colors the fundamental region based on the dynamical system model to generate periodic seamless patterns with global symmetry and controllable local details. Experiments show that the proposed method can be adapted to any quadrilateral structure, from regular rectangles to non-convex polygons. By adjusting the interpolation parameters and dynamical system functions, the symmetry, texture complexity, and visual rhythm of the patterns can be flexibly regulated. The algorithm achieves efficient computation under GPU parallel optimization (with an average generation time of 0.25 s per pattern), providing a new solution for the pattern generation and personalized design of digital art patterns.

This article introduces a novel variational approach for solving the inverse geodesic problem on a transcendental surface shaped as a cylindrical structure with a cycloidal generatrix, a type of geometry that has not been previously studied in this context. Unlike classical models that rely on symmetric surfaces such as spheres or spheroids, this method formulates the geodesic path as a functional minimization problem. By applying the Euler–Lagrange equation, an analytical integration of the corresponding second-order differential equation is achieved, resulting in a parametric expression that satisfies boundary conditions. The effectiveness of the proposed method for computing geodesic curves on transcendental surfaces has been rigorously evaluated through a series of numerical experiments. Analytical validation has been carried out using MathCad, while simulation and three-dimensional visualization have been implemented in Python. Numerical experiments are conducted and 3D visualizations of the geodesic lines are presented for multiple point pairs on the surface, demonstrating the accuracy and computational efficiency of the proposed solution. This enables a closed-form analytical representation of the geodesic curve, significantly reducing computational complexity compared to existing numerical-heuristic methods. The obtained results offer clear advantages over existing studies in the field of computational geometry and variational calculus. Specifically, the proposed method enables the construction of geodesic curves on complex transcendental surfaces where traditional methods either fail or require intensive numerical approximation. The analytical integration of geodesic equations enhances both accuracy and performance, achieving an average computational cost reduction of approximately 27-30% and accuracy improvement of around 20% in comparison with previous models utilizing non-polynomial metrics. These enhancements are especially relevant in applications requiring real-time response and precision, such as robotics, CAD systems, computer graphics, and virtual environment simulation. The method’s ability to deliver compact and exact solutions for boundary value problems positions it as a valuable contribution for both theoretical and applied sciences.

Abstract The increasing volume of data streams poses significant computational challenges for detecting changepoints online. Likelihood-based methods are effective, but a naive sequential implementation becomes impractical online due to high computational costs. We develop an online algorithm that exactly calculates the likelihood ratio test for a single changepoint in p-dimensional data streams by leveraging a fascinating connection with computational geometry. This connection straightforwardly allows us to exactly recover sparse likelihood ratio statistics: that is assuming only a subset of the dimensions are changing. Our algorithm is straightforward, fast, and apparently quasi-linear. A dyadic variant of our algorithm is provably quasi-linear, being O(nlog(n)p+1) for n data points and p less than 3, but slower in practice. These algorithms are computationally impractical when p is larger than 5, and we provide an approximate algorithm suitable for such p which is O(nplog(n)p~+1), for some user-specified p~≤5. We derive statistical guarantees for the proposed procedures in the Gaussian case, and confirm the good computational and statistical performance, and usefulness, of the algorithms on both empirical data and NBA data.

Described are neural networks that accurately reproduce proton and 13C chemical shifts obtained from ωB97X-D/6-31G*//ωB97X-D/6-31G* density functional model GIAO calculations. They support uncharged, closed-shell molecules comprising H, C, N, O, F, S, Cl, and Br. Development involved training to ≈2.7 million equilibrium geometry and chemical shift calculations for a diverse collection of organic molecules (including synthetic drugs and natural products). Referenced to ωB97X-D/6-31G*//ωB97X-D/6-31G* calculations, chemical shifts from neural networks for 601 marine natural products show RMS errors of 0.05 ppm (proton) and 0.76 ppm (13C). RMS errors of 0.09 ppm (proton) and 1.02 ppm (13C) shifts result when equilibrium geometries from a previously described "estimated ωB97X-D/6-31G*" neural network model (trained to reproduce ωB97X-D/6-31G* geometries) are utilized. A second assessment of experimental 13C chemical shifts for 246 natural products is provided. Using neural network models to provide both geometries and chemical shifts: 45% of 13C shifts reproduce experimental values within 1 ppm, 73% within 2 ppm, and 86% within 3 ppm. Utilizing neural network models for both equilibrium geometries and chemical shifts reduces the computational time required for accurate proton and 13C chemical shifts from tens to hundreds of minutes to just a few seconds per molecule.

Convex combination of points is a fundamental operation in computational geometry. By considering rigid-body displacements as points in the image spaces of planar quaternions, quaternions and dual quaternions, respectively, the notion of convexity in Euclidean three-space has been extended to kinematic convexity in , and in the context of computational kinematic geometry. This paper deals with computational kinematic geometry of bounded planar objects rather than that of infinitely large moving spaces. In this paper, we present a new formulation for kinematic convexity based on an average-distance minimizing motion sweep of a bounded planar object. The resulting 1-DOF motion sweep between two planar poses is represented as a convex combination in the configuration space defined by where is associated with the location of the centroid of the planar object and with being the angle of rotation. For three poses, a 2-DOF motion sweep is developed that not only minimizes the combined average squared distances but also attains a convex-combination representation so that existing algorithms for convex hull of points can be readily applied to the construction and analysis of kinematic convex hulls. This results in a new type of convex hull for planar kinematics such that its boundaries are defined by the average-distance minimizing sweeps of the bounded planar object.