In this paper, we study centroids, orthocenters, circumcenters, and incenters of geodesic triangles in non-Euclidean geometry, and we discuss the existence of the Euler line in this context. Moreover, we give simple proofs of the existence of a total...
An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection...
Helping learners to develop a solid grasp of geometric concepts poses a challenge for teachers. Therefore, it is important that teachers have a sound understanding of the geometry they teach. The aim of this qualitative study was to explore pre-servi...
Tarski’s first-order axiom system E2 for Euclidean geometry is notable for its completeness and decidability. However, the Pythagorean theorem—either in its modern algebraic form a2+b2=c2 or in Euclid’s Elements—cannot be dire...
This paper builds two detailed examples of generalized normal in non-Euclidean spaces, i.e., the hyperbolic and elliptic geometries. In the hyperbolic plane we define a n-sided hyperbolic polygon P, which is the Euclidean closure of the hyperbolic pl...
In this paper, an interesting symmetry in Euclidean geometry, which is broken in Lorentz geometry, is studied. As it turns out, attempting to minimize the integral of the square of the scalar curvature leads to completely different results in these t...
A cyclic antipodal pair of a circle is a pair of points that are the intersection of the circle with the diameter of the circle. In this article, a recent proof of Ptolemy’s Theorem—simultaneously in both (i) Euclidean geometry and (ii) t...
Many real-world networks can be modeled as attributed networks, where nodes are affiliated with attributes. When we implement attributed network embedding, we need to face two types of heterogeneous information, namely, structural information and att...
This work aims at investigating the geometry of surfaces corresponding to the geometry of solutions of the vortex filament equation in Euclidean 3-space E3 using the quasi-frame. In particular, we discuss some geometric properties and some characteri...
The intersection of fractals, non-Euclidean geometry, spatial autocorrelation, and urban structure offers valuable theoretical and practical application insights, which echoes the overarching goal of this paper. Its research question asks about conne...
We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping r from a two-dimensional parameter space M to the three-dimensional Euclidean space R 3 . The metric variabl...
This paper explores the performance of ChatGPT and GeoGebra Discovery when dealing with automatic geometric reasoning and discovery. The emergence of Large Language Models has attracted considerable attention in mathematics, among other fields where...
Tropical Principal Component Analysis (PCA) is an analogue of the classical PCA in the setting of tropical geometry, and applied it to visualize a set of gene trees over a space of phylogenetic trees, which is a union of lower-dimensional polyhedral...
There was an obvious parallelism between the lives of the two Hungarian men of science: János Bolyai a genial geometer and Frigyes Károlyházy a missionary of education in the field of modern physics. Both of them selected and solved a complex problem...
In this work, we study classical differential geometry of the curves according to type-2 Bishop trihedra. First, we present some characterizations of a general helix, a helix, special cases and spherical curves. Thereafter, we investigate position ve...
The non-Euclidean geometry created by Bolyai, Lobachevsky and Gauss has led to a new physical theory—general relativity. In due turn, a correct mathematical treatment of the cosmological problem in general relativity has led Friedmann to a discovery...
We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbi...
Space out of a topological defect of the Abrikosov–Nielsen–Olesen (ANO) vortex type is locally flat but non-Euclidean. If a spinor field is quantized in such a space, then a variety of quantum effects are induced in the vacuum. On the basis of the co...
The application of contemporary artificial intelligence techniques to address geometric problems and automated deductive proofs has always been a grand challenge to the interdisciplinary field of mathematics and artificial intelligence. This is the f...
In this paper, we give some classifications of the k-Yamabe solitons on the hypersurfaces of the Euclidean spaces from the vector field point of view. In several results on k-Yamabe solitons with a concurrent vector field on submanifolds in Riemannia...
We propose a geometry topological framework to analyze storm dynamics by coupling persistent homology with Anti-de Sitter (AdS)-inspired metrics. On radar images of a bow echo event, we compare Euclidean distance with three compressive AdS metrics (&...
The purpose of this paper is twofold. On a technical side, we propose an extension of the Hausdorff distance from metric spaces to spaces equipped with asymmetric distance measures. Specifically, we focus on extending it to the family of Bregman dive...
Methods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured wit...
It is well known among geometry scholars that the golden triangle, an isosceles triangle with sides and base in golden ratio, maintains a significant relationship with regular polygons, notably the regular pentagon, pentagram, and decagon. Extensive...
Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently, they have also been used extensively in artificial intelligence, in order to study (connections between) vario...
A matrix information-geometric method was developed to detect the change-points of rigid body motions. Note that the set of all rigid body motions is the special Euclidean group S E ( 3 ) , so the Riemannian mean based on the Lie group structu...
In his famous work, “Measurement of a Circle,” Archimedes described a procedure for measuring both the circumference of a circle and the area it bounds. Implicit in his work is the idea that his procedure defines these quantities. Modern...
We first briefly summarize several well-known properties of regular tessellations of the three two-dimensional maximally symmetric manifolds, E2, S2, and H2, by bounded regular tiles. For instance, there exist infinitely many regular tessellations of...
This paper investigates the properties of the Yang–Baxter equation, which was initially formulated in the field of theoretical physics and statistical mechanics. The equation’s framework is extended through Yang–Baxter systems, aimi...
A non-iterative method for the difference of means is presented to calculate the log-Euclidean distance between a symmetric positive-definite matrix and the mean matrix on the Lie group of symmetric positive-definite matrices. Although affine-invaria...
Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Ptolemy, is well-known. By means of the relativistic model of hyperbolic geometry, we translate Ptolemy’s Theorem from Euclidean geometry into the...
Ptolemy’s theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s theorem from analytic Euclidean geometry into the Poincaré ball model of analytic h...
While there are well-known synthetic methods in the literature for finding the image of a point under circular inversion in l2-normed geometry (Euclidean geometry), there is no similar synthetic method in Minkowski geometry, also known as the geometr...
We study the geometry of Euclidean instantons in loop quantum cosmology (LQC) such as those relevant for the no-boundary proposal. Confining ourselves to the simplest case of a cosmological constant in minisuperspace cosmologies, we analyze solutions...
We study the Bäcklund transformations of integrable geometric curve flows in certain geometries. These curve flows include the KdV and Camassa-Holm flows in the two-dimensional centro-equiaffine geometry, the mKdV and modified Camassa-Holm flows in t...
Background/Objectives: In computer graphics, virtual worlds are constructed and visualized through algorithmic processes. These environments are typically populated with objects defined by mathematical models, traditionally based on Euclidean geometr...
Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n ∈ N , and discover the properties that qualify these operations to the title addition despite the fac...
Scale is a fundamental concept that has attracted persistent attention in geography literature over the past several decades. However, it creates enormous confusion and frustration, particularly in the context of geographic information science, becau...
Within the framework of the third quantization, we consider the possibility that an initially recollapsing baby universe can enter a stage of near de Sitter inflation by tunnelling through a Euclidean wormhole that connects the recollapsing and infla...
Deep neural networks are vulnerable and susceptible to adversarial examples, which can induce erroneous predictions by injecting imperceptible perturbations. Transferability is a crucial property of adversarial examples, enabling effective attacks un...
The study of fuzzy geometry and its different components has grown in recent years, establishing the formal foundations for its development. This paper is devoted to addressing some metric relations in the fuzzy right triangle; in particular, a versi...
From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the 18seventies), and what are the principal first normal directions, (first studied by Kostadin Tren...
Since the discovery of quantum mechanics, the fact that the wavefunction is defined on the 3n-dimensional configuration space rather than on the 3-dimensional space has seemed uncanny to many, including Schrödinger, Lorentz, and Einstein. Even t...
In this article, we study the intersection of a finite collection of disks in Euclidean space by examining spheres of various dimensions and their poles (extreme values with respect to canonical projections) contained within the intersection’s...
This work describes the geometry behind the Smith chart, recent 3D Smith chart tool and previously reported conceptual Hyperbolic Smith chart. We present the geometrical properties of the transformations used in creating them by means of inversive ge...
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 si...
This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where...
In this work, we study developable ruled surfaces with constant mean curvature along a curve. The mean curvature of developable ruled surfaces generated by indicatrix curves is calculated. The analysis is first carried out in Euclidean three-space an...
Fractals are geometric shapes and patterns that may repeat their geometry at smaller or larger scales. It is well established that fractals can describe shapes and surfaces that cannot be represented by the classical Euclidean geometry. An eclectic s...
Elastic media with defects are considered manifold with nontrivial Riemann–Cartan geometry in the geometric theory of defects. We obtain the solution of three-dimensional Euclidean general relativity equations with an arbitrary number of linear...
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