Computational Geometry Research Articles

Counting \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 8 black holes as algebraic varieties

We calculate the helicity trace index B14 for \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 8 pure D-brane black holes using various techniques of computational algebraic geometry and find perfect agreement with the existing results in the literature. For these black holes, microstate counting is equivalent to finding the number of supersymmetric vacua of a multi-variable supersymmetric quantum mechanics which in turn is equivalent to solving a set of multi-variable polynomial equations modulo gauge symmetries. We explore four different techniques to solve a set of polynomial equations, namely Newton Polytopes, Homotopy continuation, Monodromy and Hilbert series. The first three methods rely on a mixture of symbolic and high precision numerics whereas the Hilbert series is symbolic and admit a gauge invariant analysis. Furthermore, exploiting various exchange symmetries, we show that quartic and higher order terms are absent in the potential, which if present would have spoiled the counting. Incorporating recent developments in algebraic geometry focusing on computational algorithms, we have extended the scope of one of the authors previous works [1, 2] and presented a new perspective for the black hole microstate counting problem. This further establishes the pure D-brane system as a consistent model, bringing us a step closer to \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 2 black hole microstate counting.

An Image-based method for evaluating changes in particle size and morphology distributions of aggregate materials after vibratory compaction test

Particle breakage and abrasion of aggregate materials can be induced by field compaction during construction, which adversely affects their mechanical properties. However, current compaction quality control testing does not evaluate changes in particle size distribution and particle morphology of aggregate materials due to the lack of fast and inexpensive testing methods. In this study, a modified watershed-based segmentation image analysis method and computational geometry algorithms were used to quickly quantify changes in particle size and morphology distributions of a large amount of contacting aggregate particles. The accuracy and required minimum sample size of the image analysis method were statistically evaluated. Compared to the conventional ASTM C136 aggregate sieve analysis test, the image analysis method conducted on a 40% sample size of the conventional method yielded less than 6% difference between the results, which can meet the ASTM C136 test repeatability criteria. In addition, demonstration tests were conducted to incorporate the image analysis method into a laboratory vibratory compaction test for evaluating the overall particle size and morphology changes of an aggregate material after compaction. Results indicate that the initial gradation and water content of the aggregate material can influence the particle breakage and abrasion characteristics of aggregate materials under compaction.

ETHNOMODELLING: FRACTAL GEOMETRY ON THE DOOR ORNAMENT OF THE SUMENEP PALACE USING THE LINDENMAYER SYSTEM

The Sumenep Palace area is one of the cultural heritage relics of the Duke of Sumenep, which is still preserved today. This research aims to reveal the fractal model of door ornament in the Koneng Office of Sumenep Palace. A qualitative method with an ethnography type is used. Data were collected through initial observation, literature study, interviews, and documentation. Analysis of the fractal form is done with the Lindenmayer system method and constructed with the help of the L-Studio application. The results of the study revealed the existence of fractal structures on the door ornament of the Sumenep palace office. The length, angle, and ratio of each part of the ornament influence the fractal shape. The findings of this study can be utilized in learning mathematics in lectures, such as fractal geometry and computational geometry.

PSI-21 Optimizing sample size for accurate measurements and prediction equations for intestinal morphology in broiler chickens

Abstract Histological measurements of intestinal morphology, including villi height (VH) and crypt depth (CD), are routinely used to evaluate intestinal health and the nutrient absorptive capacity in monogastric animals. These metrics suffer from significant natural variation, and there is no established standard for the number of unique measurements required to robustly estimate these variables for a given animal. Furthermore, the potential utility of additional variables, including villus width (VW), perimeter (VP), or villus area (VA), has not been explored. Therefore, the objective of the current experiment was to estimate the minimum number of measurements required to control the intrasample variance in each variable and determine if more complex or time-consuming metrics, such as VP and/or VA, can be accurately estimated from linear measurements. A set of 1283 intestinal morphology data, including VH, CD, VW, VP, and VA, measured from 68 intestinal tissues (46 ileum and 22 jejunum) of 4 experiments were used in the current study (Figure 1). Retrospective calculation of the CV for all combinations of N measurements per sample was used to determine the impact of measurement number on the quality of data. Data for the jejunum and ileum were pooled together, and the NLIN procedure of SAS was used to estimate the optimal measurement for intestinal morphology measurements using a quadratic broken line. The VA was estimated using geometric calculation with the assumption that VA equals 0.67 × base width × VH. A paired t-test was conducted to compare measured and calculated VA. The optimal number of measurements to minimize within-sample CV for VH, CD, VW, VP, and VA ranged from 10.6 to 10.8. The VH was highly correlated (P < 0.05) with VP (r = 0.95 and 0.87) and VA (r = 0.77 and 0.63) in the jejunum and ileum, respectively (Table 1 and 2). Additionally, there was a significant difference (P < 0.05) between measured (0.110 µm2) and calculated VA (0.086 µm2). In conclusion, this study shows that measuring 11 villi per sample is required for minimizing variations within the jejunal or ileal tissue in broiler chickens. Also, estimating VA based on idealized geometry was found to be inaccurate, necessitating either direct measurement or more complex prediction equations.

Perceptually Inspired C0-Continuity Haptic Shape Display with Trichamber Soft Actuators.

Shape display devices composed of actuation pixels enable dynamic rendering of surface morphological features, which have important roles in virtual reality and metaverse applications. The traditional pin-array solution produces sidestep-like structures between neighboring pins and normally relies on high-density pins to obtain curved surfaces. It remains a challenge to achieve continuous curved surfaces using a small number of actuated units. To address the challenge, we resort to the concept of surface continuity in computational geometry and develop a C0-continuity shape display device with trichamber fiber-reinforced soft actuators. Each trichamber unit produces three-dimensional (3D) deformation consisting of elongation, pitch, and yaw rotation, thus ensuring rendered surface continuity using low-resolution actuation units. Inspired by human tactile discrimination threshold on height and angle gradients between adjacent units, we proposed the mathematical criteria of C0-continuity shape display and compared the maximal number of distinguishable shapes using the proposed device in comparison with typical pin-array. We then established a shape control model considering the nonlinearity of soft materials to characterize and control the soft device to display C0-continuity shapes. Experimental results showed that the proposed device with nine trichamber units could render typical sets of distinguishable C0-continuity shape sequence changes. We envision that the concept of C0-continuity shape display with 3D deformation capability could improve the fidelity of the rendered shapes in many metaverse scenarios such as touching human organs in medical palpation simulations.

Construction of the Ellipse with Maximum Area Inscribed in an Arbitrary Convex Quadrilateral

An ellipse can be uniquely determined by five tangents. Given a convex quadrilateral, there are infinitely many ellipses inscribed in it, but the one with maximum area is unique. In this paper, we give a concise and effective solution of this problem. Our solution is composed of three steps: First, we transform the problem from the maximal ellipse construction problem into the minimal quadrilateral construction problem by an affine transformation. And then, we convert the construction problem into a conditional extremum problem by analyzing the key angles. At last, we derive the solution of the conditional extremum problem with Lagrangian multiplier. Based on the conclusion, we designed an algorithm to achieve the construction. The numerical experiment shows that the ellipse constructed by our algorithm has the maximum area. It is interesting and surprising that our constructions only need to solve quadratic equations, which means the geometric information of the ellipse can even be derived with ruler and compass constructions. The solution of this problem means all the construction problems of conic with extremum area from given pure tangents are solved, which is a necessary step to solve more problems of constructing ellipses with extremum areas. Our work also provides a useful conclusion to solve the maximal inscribed ellipse problem for an arbitrary polygon in Computational Geometry.

I❤MESH: A DSL for Mesh Processing

I❤MESH: A DSL for Mesh Processing

Non‐Euclidean Sliced Optimal Transport Sampling

AbstractIn machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well‐dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real‐world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space 𝕊d, the hyperbolic plane ℍd, and the real projective plane ℙd. Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.

FOUNDATION AND ASSESSMENT OF OPTIMAL FRACTURE GEOMETRY IN MULTI-STAGE HYDRAULIC FRACTURING OF HORIZONTAL WELLS

The article is dedicated to the calculation and assessment of the optimal geometry of fractures in the context of conducting hydraulic fracturing in horizontal wells. The significance of correctly determining the geometry of fractures to maximize well productivity and the efficiency of the hydraulic fracturing process is considered. The paper analyzes various methods for calculating and evaluating the optimal geometry of fractures, including numerical modeling, experimental research, and practical observations.Additionally, the article covers the results of experimental studies aimed at analyzing the behavior of fractures in different geological conditions. These studies include the analysis of physical and mechanical properties of rock formations, as well as the influence of well parameters on fracture formation. The obtained results can be useful for engineers and specialists in the oil and gas industry in designing and conducting hydraulic fracturing operations to increase hydrocarbon recovery from reservoirs.

Collaborative CAD Technology and Computational Geometry Algorithm in Digital Building Skin Design

Collaborative CAD Technology and Computational Geometry Algorithm in Digital Building Skin Design

An Algorithm for Distracted Driving Recognition Based on Pose Features and an Improved KNN

To reduce safety accidents caused by distracted driving and address issues such as low recognition accuracy and deployment difficulties in current algorithms for distracted behavior detection, this paper proposes an algorithm that utilizes an improved KNN for classifying driver posture features to predict distracted driving behavior. Firstly, the number of channels in the Lightweight OpenPose network is pruned to predict and output the coordinates of key points in the upper body of the driver. Secondly, based on the principles of ergonomics, driving behavior features are modeled, and a set of five-dimensional feature values are obtained through geometric calculations. Finally, considering the relationship between the distance between samples and the number of samples, this paper proposes an adjustable distance-weighted KNN algorithm (ADW-KNN), which is used for classification and prediction. The experimental results show that the proposed algorithm achieved a recognition rate of 94.04% for distracted driving behavior on the public dataset SFD3, with a speed of up to 50FPS, superior to mainstream deep learning algorithms in terms of accuracy and speed. The superiority of ADW-KNN was further verified through experiments on other public datasets.

Sequential sampling for functional estimation via Sieve

AbstractSequential sampling methods are often used to estimate functions describing models subjected to time‐intensive simulations or expensive experiments. These methods provide guidelines for point selection in the domain to capture maximum information about the function. However, in most sequential sampling methods, determining a new point is a time‐consuming process. In this paper, we propose a new method, named Sieve, to sequentially select points of an initially unknown function based on the definition of proper intervals. In contrast with existing methods, Sieve does not involve function estimation at each iteration. Therefore, it presents a greater computational efficiency for achieving a given accuracy in estimation. Sieve brings in tools from computational geometry to subdivide regions of the domain efficiently. Further, we validate our proposed method through numerical simulations and two case studies on the calibration of internal combustion engines and the optimal exploration of an unknown environment by a mobile robot.

Application of Novel Ruthenium (II) Polypyridyl Complexes as Robust DNA Probes, Optical Material and Antimicrobials-An Experimental and DFT Approach.

The nature of the interaction of DNA with heteroleptic Ruthenium (II) Polypyridyl complexes of the type [Ru (A)2TPIP]2+, where TPIP = 2-(1-p-tolyl-1H pyrazol-4 -yl)-1H-imidazo [4, 5-f[1. 10] phenanthroline and A = 1,10 phenanthroline (1),4,4'-dimethyl-1,10-ortho Phenanthroline (2),2,2' - bipyridine(3) and 4, 4' dimethyl 2, 2'- bipyridine (4), has been investigated by experimentaland molecular docking approaches. The order of the DNA binding affinities of the synthesised complexes is 1 > 2 > 3 > 4. The findings imply that the unsubstitutedcomplex has a better affinity to bind with DNA than the substituted (dmp and dmb) emphasizing the significance of the auxiliary ligand. Additionally, as the medium's ionic strength drops, the DNA/Ru ratio rises, or when water is displaced by glycerol, the intercalation of complexes into DNA increases. DFT calculations at the B3LYP/LANL2MBlevel was used for molecular geometry (Ground State) and electronic characteristic calculations. The HOMO-LUMO gap of the Ru [II]complex is less than the intercalator and hence kinetically labile. Among the complexes, the bpy complex has shown utmost non-linear optical properties (α = -153.9099 10-24esuand β = 3.8498 10-30esu). The docking study shows the significance of the Metal-intercalator's shorter length may increase DNA binding affinity. This study divulges that the Ruthenium (II) polypyridyl complexes bind to DNA preponderantly by intercalation supporting Viscosity studies. All the complexes have a considerable attraction for guanine. The standarddiskdiffusion method reveals that complexes (1,2,3 and 4) have good antibacterial activity.

Effect of bending deformation on the electronic and optical properties of O atoms adsorbed by Be3N2.

In this paper, the optimum coverage of 4.44% and the optimum adsorption sites were determined for the Be3N2 adsorption system of O atoms at different coverages based on density functional theory. The electronic and optical properties of the model were investigated by applying bending deformation to the model at these coverage and adsorption sites. Adsorption of O atoms disrupts the geometrical symmetry of Be3N2, resulting in orbital rehybridization and lowering its band gap. Bending deformation causes the band gap of the adsorbed O atom structure of Be3N2 to first increase and then decrease, resulting in the modulation of its band gap. With increasing bending deformation, the adsorbed system is redshifts, and the degree of redshift increases with increasing bending deformation. All calculations in this paper were performed using the first-principles-based CASTEP module of Materials Studio (MS). The generalized gradient approximation (GGA) plane-wave pseudopotential method and the Perdew-Burke-Ernzerhof (PBE) Perdew et al. Phys Rev Lett 77:3865, 1996 generalized functional were used in the geometry optimization and calculation process to calculate the exchange-correlation potential between electrons. The effect of coverage on the electronic and optical properties of the Be3N2-adsorbed O atom system was investigated by adsorbing different numbers of O atoms on a monolayer of Be3N2. The Be3N2 protocell contains two N atoms and three Be atoms with a space community of P6/MMM (No.191). The original cell was expanded 3 times along the direction of the base vectors a and b in the Be3N2 plane to create a 3 × 3 × 1 monolayer Be3N2 supercell system. A vacuum layer of 15Å is set in the direction of the crystal plane of the vertical monolayer Be3N2 supercell to eliminate interactions between adjacent layers. In the overall energy convergence test of the Be3N2 supercell, the plane wave truncation energy was set to 500eV, and the energy difference between the calculations given in the literature Reyes-Serrato et al. J Phys Chem Solids 59:743-6, 1998 using 550eV was less than 0.01eV, verifying the reliability of the data at a truncation energy of 500eV. The Monkhorst-Pack special k-point sampling method Monkhorst et al. Phys Rev B 13:5188, 1976 was used in the structural calculations, and the grid was set to 3 × 3 × 1. The geometric optimization parameters are set as follows: the self-consistent field iteration convergence criterion is 2.0 × 10-6eV, and the iterative accuracy convergence value is not less than 1.0 × 10-5eV/atom for the total force of each atom and less than 0.03eV/Å for all atomic forces. In addition the high-symmetry k-point path is taken as Γ(0,0,0) → M(0,0.5,0) → K(- 1/3,2/3,0) → Γ(0,0,0) Chen et al, AIP Adv 8:105105, 2018.

Innovative Design of Product Modeling Based on Computational Geometry and Multimedia Technology

Innovative Design of Product Modeling Based on Computational Geometry and Multimedia Technology

Automatic Generation and Optimization of Product Shapes Based on Generative Artificial Intelligence (AIGC) and Computational Geometry Algorithms

Automatic Generation and Optimization of Product Shapes Based on Generative Artificial Intelligence (AIGC) and Computational Geometry Algorithms

Hwang-Oguiso invariants and frozen singularities in special geometry

In Special Geometry there are two inequivalent notions of “Kodaira type” for a singular fiber: one associated with its local monodromy and one with its Hwang-Oguiso characteristic cycle. When the two Kodaira types are not equal the geometry is subtler and its deformation space gets smaller (“partially frozen” singularities). The paper analyzes the physical interpretation of the Hwang-Oguiso invariant in the context of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 QFT and describes the surprising phenomena which appear when it does not coincide with the monodromy type. The Hwang-Oguiso multiple fibers are in one-to-one correspondence with the partially frozen singularities in M-theory compactified on a local elliptic K3: a chain of string dualities relates the two geometric set-ups. Paying attention to a few subtleties, this correspondence explains in purely geometric terms how the “same” Kodaira elliptic fiber may have different deformations spaces. The geometric computation of the number of deformations agrees with the physical expectations. At the end we briefly outline the implications of the Hwang-Oguiso invariants for the classification program of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs.

Chaos in Stochastic 2d Galerkin-Navier–Stokes

We prove that all Galerkin truncations of the 2d stochastic Navier–Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies N≥392\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 392$$\\end{document}. By “chaotic” we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degeneracy of a matrix Lie algebra implying Hörmander’s condition for the Markov process lifted to the sphere bundle (projective hypoellipticity). The purpose of this work is to reformulate this condition to be more amenable for Galerkin truncations of PDEs and then to verify this condition using (a) a reduction to genericity properties of a diagonal sub-algebra inspired by the root space decomposition of semi-simple Lie algebras and (b) computational algebraic geometry executed by Maple in exact rational arithmetic. Note that even though we use a computer assisted proof, the result is valid for all aspect ratios and all sufficiently high dimensional truncations; in fact, certain steps simplify in the formal infinite dimensional limit.

Quantum secure multi-party computational geometry based on multi-party summation and multiplication

Secure multi-party computational geometry is a branch of secure multi-party computation, which is applied in many important fields. But up to now, the research on how to solve this problem with quantum methods has just started. Therefore, we study the design of quantum secure multi-party computational geometry (QSMCG) protocols in this paper. As the foundation, we extend the two-party summation and multiplication protocols proposed by Sutradhar and Om to multi-party protocols. Then we propose three new QSMCG protocols. Our proposed quantum secure two-party distance computation protocol has improvement in efficiency and accuracy. And the proposed quantum secure multi-party polytope volume computation protocol as well as the quantum secure circle–circle intersection area computation protocol are designed for the first time. Moreover, these protocols are designed to use multi-third parties, which solves the problem of intermediate value leakage in computation. The correctness, security and performance of all these protocols are analyzed.

A methodology for applying isogeometric inverse finite element method to the shape sensing of stiffened thin-shell structures

The capability of reconstructing the displacement field of a structure based on in-situ strain measurements is referred to as “shape sensing” and constitutes a fundamental unit for the real-time monitoring of critical structural components. The Inverse Finite Element Method (iFEM) is one of the most promising innovative techniques for accomplishing this task. This study explores the application of iFEM in conjunction with Isogeometric Analysis (IGA) for the shape sensing of stiffened thin-shell structures. The use of IGA allows for the exact representation of computational geometry, simplifies mesh refinement, and potentially reduces the number of installed sensors. In the context of IGA, when the geometry of a structure is composed of multiple surfaces, non-conforming interfaces are typically involved. To overcome this issue, we propose an assembly process based on the Lagrange multiplier method to jointly apply iFEM and IGA to shell structures assemblies. The methodology is numerically validated using FEM models both to generate the in-situ strain data and as a comparison for iFEM reconstruction results. The case studies include a T-beam, a reinforced hyperbolic paraboloid, and a wingbox. Convergence analyzes are performed to investigate the shape sensing accuracy as the number of inverse elements increases. Additionally, contour plots of displacement field components are compared with the reference solution, revealing a high degree of agreement.