Article
Mathematics, Applied
Alaa Eddine Bensad, Aziz Ikemakhen
Summary: We propose a general method for defining and efficiently computing barycentric coordinates with respect to polygons on the unit sphere. We develop a novel explicit construction to compute the spherical barycentric coordinates from their 2D-Euclidean counterparts and provide families of spherical coordinates for convex and non-convex spherical polygons. We also present an alternative construction for spherical barycentric coordinates using 3D barycentric coordinates for closed triangular meshes, which can be extended to arbitrary dimensions. Our spherical and 3D coordinates have wide applicability in various domains, demonstrated through examples in spherical blending, space deformations, and shape morphing in 3D.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Changcheng Wei, Muhammad Salman, Syed Shahzaib, Masood Ur Rehman, Juanyan Fang
Summary: The edge metric dimension of a graph refers to the number of vertices in the smallest set X that can distinguish every two edges in the graph. In this article, the authors solve the edge metric dimension problem for certain classes of planar graphs.
Article
Engineering, Electrical & Electronic
Yingiu Xia, Chengpu Yu, Chengyang He
Summary: This paper studies the problem of exploratory distributed localization of networked mobile agents in a GPS-Denied 3D environment. The authors provide an analytic solution and develop a distributed algorithm to solve the static and mobile target localization problems. The proposed method exhibits good scalability and effectiveness in different scenarios.
IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER NETWORKS
(2022)
Article
Computer Science, Software Engineering
Alaa Eddine Bensad, Aziz Ikemakhen
Summary: This paper presents the construction of hyperbolic Barycentric coordinates on the hyperbolic plane, including hyperbolic Wachspress, mean values, and discrete harmonic coordinates. These coordinates are unique for points in a hyperbolic triangle and are derived on the Poincare disk model. Furthermore, the paper demonstrates the applications of hyperbolic parameterization, such as hyperbolic deformation and shape morphing.
COMPUTER AIDED GEOMETRIC DESIGN
(2022)
Article
Mathematics
Odysseas Kosmas, Pieter Boom, Andrey P. Jivkov
Summary: This article discusses the deformation of solids due to changing boundary conditions, presenting a discrete energy model based on mappings between nodal positions and deformation gradient invariants. The analysis of these deformations is achieved through energy minimization, with constraints eliminated using Lagrange multipliers. The accuracy of the technique is verified through numerical examples, demonstrating its potential in describing solid deformation.
Article
Mathematics
Shaheen Nazir
Summary: This paper studies two important subdivisions, namely Cheeger-Muller-Schrader's subdivision and the r-colored barycentric subdivision, and proves that these subdivisions have the same f-vector. By analyzing the transformation matrices of the f- and h-vectors, the paper provides an explicit description of the relationship between these subdivisions and the original poset. This research is significant for describing and understanding posets of multichains.
DISCRETE MATHEMATICS
(2023)
Article
Education, Scientific Disciplines
U-Rae Kim, Wooyong Han, Dong-Won Jung, Jungil Lee, Chaehyun Yu
Summary: The article computes the electrostatic potential of a uniformly charged triangle using barycentric coordinates, resulting in good agreement with numerical results. It investigates the asymptotic behavior of the analytic expression in special limits and provides useful expressions for improving numerical convergence at boundaries. Appendices include integral tables, parametrization of the gradient operator, and coordinate-transformation rules between barycentric and Cartesian coordinates.
EUROPEAN JOURNAL OF PHYSICS
(2021)
Article
Thermodynamics
Thorsten Zirwes, Feichi Zhang, Yiqing Wang, Peter Habisreuther, Jordan A. Denev, Zheng Chen, Henning Bockhorn, Dimosthenis Trimis
Summary: This study introduces a tracking algorithm for Flame Particles (FP) using barycentric coordinates, which adds value in studying local flame dynamics. By seeding FPs along the flame surface and tracking their trajectories at arbitrary points on the flame front, a phase shift between the unsteady flame stretch rate and local flame speed is revealed.
PROCEEDINGS OF THE COMBUSTION INSTITUTE
(2021)
Article
Mathematics
Carlos Ortiz, Adriana Lara, Jesus Gonzalez, Ayse Borat
Summary: The algorithm described and implemented in this paper produces an explicit system of piecewise linear motion planners for an automated guided vehicle by inputting a polyhedron. The cardinality of the output is probabilistically close to the minimal possible cardinality, providing an automated solution for robust robot motion planning. The implementation includes discretizing the homotopic distance concept and computer estimations of other invariants, such as the Lusternik-Schnirelmann category of polyhedra.
Letter
Education, Scientific Disciplines
U-Rae Kim, Wooyong Han, Dong-Won Jung, Jungil Lee, Chaehyun Yu
Summary: Researchers conducted a comprehensive numerical comparison of three known analytic expressions for the electrostatic potential of a uniformly charged triangle sheet, and found that the three expressions are equivalent to each other.
EUROPEAN JOURNAL OF PHYSICS
(2021)
Article
Physics, Multidisciplinary
U-Rae Kim, Dong-Won Jung, Chaehyun Yu, Wooyong Han, Jungil Lee
Summary: This study evaluates the inertia tensor of a triangular plate of uniform mass distribution using the barycentric coordinate system, expressing physical quantities in terms of a single master integral and employing Lagrange undetermined multipliers for faster computation. The moment of inertia is uniquely expressed in terms of mass, barycentric coordinates of the pivot, and side lengths, with the most compact expression in comparison to commonly used ones in mechanical engineering. Appendices provide necessary master integrals for computing integrals over triangles in the barycentric coordinate system and derivations of barycentric coordinates of common triangle centers. The barycentric coordinates are expected to be efficient in computing physical quantities like the electrostatic potential of a triangular charge distribution and can be applied to general-physics experiments through practical experimental designs.
JOURNAL OF THE KOREAN PHYSICAL SOCIETY
(2021)
Article
Mathematics
Samsul Ariffin Abdul Karim, Faheem Khan, Ghulam Mustafa, Aamir Shahzad, Muhammad Asghar
Summary: Subdivision schemes use rules to transform polygons into smooth curves or surfaces. Determining the number of iterations needed to achieve the desired shape with a user-specified error tolerance is a challenge. This paper presents a new approach based on convolution to estimate error bounds and subdivision depth for non-stationary schemes.
Review
Geography, Physical
Hefeng Liu, Yi Fang
Summary: The paper proposes a direct 3D coordinate transformation method based on the affine invariance of barycentric coordinates, which avoids solving transformation parameters, requires neither initial values nor iterations. Experiments show that the proposed method can reach or even outperform traditional methods in LiDAR point cloud registration and coordinate transformation.
JOURNAL OF SPATIAL SCIENCE
(2021)
Article
Computer Science, Artificial Intelligence
Weiyue Zhao, Hao Lu, Zhiguo Cao, Xin Li
Summary: Geometric-invariant coordinate representations, such as Degree, can significantly reduce mismatches between features and improve matching quality. Degree, a novel anchor-to-barycentric (A2B) coordinate encoding approach, shows competitive performance in solving the problem of repeated patterns and achieving state-of-the-art feature correspondences.
INTERNATIONAL JOURNAL OF COMPUTER VISION
(2023)
Article
Mathematics, Applied
Clemens Hofreither
Summary: The algorithm, named BRASIL, iteratively adjusts interval lengths to achieve the best rational approximation of a scalar function by exploiting the equioscillation of local maximum errors. It utilizes the barycentric rational formula for stable computation and demonstrates improved convergence rate with the Anderson acceleration method. The algorithm shows excellent numerical stability and computational efficiency for high degree rational approximations.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Emiliano Cirillo, Kai Hormann
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Chongyang Deng, Huixia Xu, Weiyin Ma, Yajuan Li
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Dmitry Anisimov, Kai Hormann, Teseo Schneider
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Emiliano Cirillo, Kai Hormann, Jean Sidon
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2020)
Article
Computer Science, Software Engineering
Kai Hormann, Jianmin Zheng
COMPUTER AIDED GEOMETRIC DESIGN
(2020)
Article
Computer Science, Software Engineering
Chongyang Deng, Qingjun Chang, Kai Hormann
COMPUTER AIDED GEOMETRIC DESIGN
(2020)
Article
Computer Science, Software Engineering
Jin Xie, Jinlan Xu, Zhenyu Dong, Gang Xu, Chongyang Deng, Bernard Mourrain, Yongjie Jessica Zhang
COMPUTER AIDED GEOMETRIC DESIGN
(2020)
Article
Computer Science, Software Engineering
Rida T. Farouki, Kai Hormann, Federico Nudo
COMPUTER AIDED GEOMETRIC DESIGN
(2020)
Article
Computer Science, Software Engineering
C. Gotsman, K. Hormann
Summary: The study focuses on the landmark distance function in a simply connected planar polygon, demonstrating the effectiveness of steepest descent in generating paths between any two points in the polygon without getting stuck.
COMPUTER GRAPHICS FORUM
(2021)
Article
Computer Science, Software Engineering
Nira Dyn, Kai Hormann, Claudio Mancinelli
Summary: This paper explores subdivision schemes, focusing on univariate, linear, binary schemes, as well as deriving and applying properties of non-stationary and non-uniform schemes.
COMPUTER AIDED GEOMETRIC DESIGN
(2022)
Article
Computer Science, Software Engineering
Zhihao Wang, Yajuan Li, Huixia Xu, Jianzhen Liu, Chongyang Deng
Summary: This paper introduces a new parametric spline curve, called P-spline curve, and discusses its definition and construction. P-spline curves have the advantages of adjustable continuous orders and local influences, as well as simple construction and intuitive relations.
Article
Computer Science, Software Engineering
Kaikai Qin, Yajuan Li, Chongyang Deng
Summary: This study introduces a new n-sided control point-based surface patch called the blending Bezier patch (BB patch). It is constructed by creating corner Bezier surfaces and using Gregory corner blending. The BB patch is defined on a regular polygonal domain and can be joined easily to surrounding Bezier and other BB patches due to its similarity in boundary behaviors with the Bezier patch. A practical application of the BB patch is filling holes with G2 continuity.
COMPUTER AIDED GEOMETRIC DESIGN
(2023)
Article
Computer Science, Software Engineering
Qingjun Chang, Weiyin Ma, Chongyang Deng
Summary: In this paper, we propose a constrained least square progressive and iterative approximation (CLSPIA) method to solve the problem of B-spline curve and surface fitting with constraint on data interpolation. The CLSPIA method inherits all the nice properties of LSPIA, and it is efficient and effective.
Proceedings Paper
Computer Science, Artificial Intelligence
Craig Gotsman, Kai Hormann
Summary: The paper proposes a novel method for compact storage of geodesic distance information and efficient point-to-point geodesic distance queries. It achieves this through a nested bisection of the mesh surface and compactly describing distances between mesh vertices and relevant subset of curves. The method provides a good tradeoff between database size, query runtime, and result accuracy.
PROCEEDINGS SIGGRAPH ASIA 2022
(2022)