Search Results (135)

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 connections between graph theory adjacency matrix eigenfunctions and certain non-Euclidean grid systems; its explorations reflect accompanying synergistic influences on modern urban design. A Minkowski metric with an exponent between one and two bridges Manhattan and Euclidean spaces, supplying an effective tool in these pursuits. This model coalesces with urban fractal dimensions, shedding light on network density and human activity compression. Unlike Euclidean geometry, which assumes unique shortest paths, Manhattan geometry better represents human movements that typically follow multiple equal-length network routes instead of unfettered straight-line paths. Applying these concepts to urban spatial models, like the Burgess concentric ring conceptualization, reinforces the need for fractal analyses in urban studies. Incorporating a fractal perspective into eigenvector methods, particularly those affiliated with spatial autocorrelation, provides a deeper understanding of urban structure and dynamics, enlightening scholars about city evolution and functions. This approach enhances geometric understanding of city layouts and human behavior, offering insights into urban planning, network density, and human activity flows. Blending theoretical and applied concepts renders a clearer picture of the complex patterns shaping urban spaces. Full article

This article describes procedures and thoughts regarding the reconstruction of geometry-given data and its uncertainty. The data are considered as a continuous fuzzy point cloud, instead of a discrete point cloud. Shape fitting is commonly performed by minimizing the discrete Euclidean distance; however, we propose the novel approach of using the expected Mahalanobis distance. The primary benefit is that it takes both the different magnitude and orientation of uncertainty for each data point into account. We illustrate the approach with laser scanning data of a cylinder and compare its performance with that of the conventional least squares method with and without random sample consensus (RANSAC). Our proposed method fits the geometry more accurately, albeit generally with greater uncertainty, and shows promise for geometry reconstruction with laser-scanned data. Full article

This study examines the spinor representations of $\mathbf{TN}$ (tangent and normal), $\mathbf{NB}$ (normal and binormal), $\mathbf{TB}$ (tangent and binormal) and $\mathbf{TNB}$ (tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space ${\mathbb{E}}^3$. Spinors are complex column vectors and move on Pauli spin matrices. Isotropic vectors in the ${\mathbb{C}}^3$ complex vector space form a two-dimensional surface in the ${\mathbb{C}}^2$ complex space. Additionally, each isotropic vector in ${\mathbb{C}}^3$ space corresponds to two vectors in ${\mathbb{C}}^2$ space, called spinors. Based on this information, our goal is to establish a relationship between curve theory in differential geometry and spinor space by matching a spinor with an isotropic vector and a real vector generated from the vectors of the Frenet–Serret frame of a curve in three-dimensional Euclidean space. Accordingly, we initially assume two spinors corresponding to the Frenet–Serret frames of the main curve and its ($\mathbf{TN}$, $\mathbf{NB}$, $\mathbf{TB}$ and $\mathbf{TNB}$)–Smarandache curves. Then, we utilize the relationships between the Frenet frames of these curves to examine the connections between the two spinors corresponding to these curves. Thus, we give the relationships between spinors corresponding to these Smarandache curves. For this reason, this study creates a bridge between mathematics and physics. This study can also serve as a reference for new studies in geometry and physics as a geometric interpretation of a physical expression. Full article

The spectral reflectance measured in situ is often regarded as the “truth”. However, its limited coverage and large spatial heterogeneity often make the ground-based reflectance unable to represent the remote sensing images. Since the spatial scale mismatch between ground-based, airborne, and spaceborne measurements, the applications of geological exploration, metallogenic prognosis and mine monitoring are facing severe challenges. In order to explore the influence of spatial scale effect on rock spectra, spectral reflectance with uncertainty caused by differences in illumination view geometry and spatial heterogeneity is introduced into the Bayesian Maximum Entropy (BME) method. Then, the rock spectra are upscaled from the point-scale to meter-scale and to 10 m-scale, respectively. Finally, the influence of spatial scale effect is evaluated based on the reflectance value, spectral shape, and spectral characteristic parameters. The results indicate that the BME model shows better upscaling accuracy and stability than Ordinary Kriging and Ordinary Least Squares model. The maximum Euclidean Distance of rock spectra caused by spatial resolution change is 6.271, and the Spectral Angle Mapper can reach 0.370. The spectral absorption position, absorption depth, and spectral absorption index are less affected by scale effect. For the area with similar spatial heterogeneity to the Huangshan Copper–Nickel Ore District, when the spatial resolution of the image is greater than 10 m, the rock’s spectrum is less influenced by the change in spatial resolution. Otherwise, the influence of spatial scale effect should be considered in applications. In addition, this work puts forward a set of processes to evaluate the influence of spatial scale effect in the study area and carry out the upscaling. Full article

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, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in $2+1$ dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras. Full article

Point cloud registration is one of the fundamental tasks in computer vision, but faces challenges under low overlap conditions. Recent approaches use transformers and overlapping masks to improve perception, but mask learning only considers Euclidean distances between features, ignores mismatches caused by fuzzy geometric structures, and is often computationally inefficient. To address these issues, we introduce a novel matching framework. Firstly, we fuse adaptive graph convolution with PPF features to obtain rich feature perception. Subsequently, we construct a PGT framework that uses GeoTransformer and combines it with location information encoding to enhance the geometry perception between source and target clouds. In addition, we improve the visibility of overlapping regions through information exchange and the AIS module, aiming at subsequent keypoint extraction, preserving points with distinct geometrical structures while suppressing the influence of non-overlapping regions to improve computational efficiency. Finally, the mask is refined through contrast learning to preserve geometric and distance similarity, which helps to compute the transformation parameters more accurately. We have conducted comprehensive experiments on synthetic and real-world scene datasets, demonstrating superior registration performance compared to recent deep learning methods. Our approach shows remarkable improvements of 68.21% in $R_{RMSE}$ and 76.31% in $t_{RMSE}$ on synthetic data, while also excelling in real-world scenarios with enhancements of 76.46% in $R_{RMSE}$ and 45.16% in $t_{RMSE}$. Full article

Since the discovery of quantum mechanics, the fact that the wavefunction is defined on the $3\mathbf{n}$-dimensional configuration space rather than on the 3-dimensional space has seemed uncanny to many, including Schrödinger, Lorentz, and Einstein. Even today, this continues to be seen as a significant issue in the foundations of quantum mechanics. In this article, it will be shown that the wavefunction is, in fact, a genuine object on space. While this may seem surprising, the wavefunction does not possess qualitatively new features that were not previously encountered in objects known from Euclidean geometry and classical physics. The methodology used involves finding equivalent reinterpretations of the wavefunction exclusively in terms of objects from the geometry of space. The result is that we will find the wavefunction to be equivalent to geometric objects on space in the same way as was always the case in geometry and physics. This will be demonstrated to hold true from the perspective of Euclidean geometry, but also within Felix Klein’s Erlangen Program, which naturally fits into the classification of quantum particles by the representations of spacetime isometries, as realized by Wigner and Bargmann, adding another layer of confirmation. These results lead to clarifications in the debates about the ontology of the wavefunction. From an empirical perspective, we already take for granted that all quantum experiments take place in space. I suggest that the reason why this works is that they can be interpreted naturally and consistently with the results presented here, showing that the wavefunction is an object on space. Full article

In this paper, the authors generalize the fractional curvature of plane curves introduced by Rubio et al. in 2023, to regular curves in the Euclidean space ${\mathbb{R}}^3$, and study the geometric properties of the curve using Caputo’s fractional derivative. Furthermore, we introduce a new definition of fractional curvature and fractional mean curvature of a regular surface, using fractional principal curvatures; and prove that such concepts are invariant under isometries; i.e., they belong to the intrinsic geometry of the regular surface. Also, a geometric interpretation is given to Caputo’s fractional derivative of algebraic polynomials. Full article

The estimation of vessels’ centerlines is a critical step in assessing the geometry of the vessel, the topological representation of the vessel tree, and vascular network visualization. In this research, we present a novel method for obtaining geometric parameters from peripheral arteries in 3D medical binary volumes. Our approach focuses on centerline extraction, which yields smooth and robust results. The procedure starts with a segmented 3D binary volume, from which a distance map is generated using the Euclidean distance transform. Subsequently, a skeleton is extracted, and seed points and endpoints are identified. A search methodology is used to derive the best path on the skeletonized 3D binary array while tracking from the goal points to the seed point. We use the distance transform to calculate the distance between voxels and the nearest vessel surface, while also addressing bifurcations when vessels divide into multiple branches. The proposed method was evaluated on 22 real cases and 10 synthetically generated vessels. We compared our method to different state-of-the-art approaches and demonstrated its better performance. The proposed method achieved an average error of 1.382 mm with real patient data and 0.571 mm with synthetic data, both of which are lower than the errors obtained by other state-of-the-art methodologies. This extraction of the centerline facilitates the estimation of multiple geometric parameters of vessels, including radius, curvature, and length. Full article

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 totally geodesic 2-dimensional submanifold containing a given geodesic triangle in the hyperbolic or spherical 3-dimensional geometry. Full article

In the Euclidean form of the theory of gravity, where there is no dedicated time parameter, a generalized canonical form of the principle of least action is proposed. On its basis, the quantum principle of least action is formulated, in which the “dynamics” of the universe in the Origin is described by the eigenvector of the action operator—the wave functional on the space of 4D Riemannian geometries and configurations of matter fields in some compact region of 4D space. The corresponding eigenvalue of the action operator determines the initial state for the world history of the universe outside this region, where the metric signature is Lorentzian and, thus, the time parameter exists. The boundary of the Origin region is determined by the condition that the rate of change of the determinant of the 3D metric tensor is zero on it. The size of the Origin is interpreted as a reciprocal temperature of the universe in the initial state. It has been suggested that in the initial state, the universe contains a certain distribution of its own mass, which is not directly related to the fields of matter. Full article

In this paper, we introduce an invariant by image viewpoint changes by applying an important theorem in conformal geometry stating that every surface of the Minkowski space ${\mathbb{R}}^{3,1}$ leads to an invariant by conformal transformations. For this, we identify the domain of an image to the disjoint union of horospheres ${\coprod }_{\alpha }{H}_{\alpha }$ of ${\mathbb{R}}^{3,1}$ by means of the powerful tools of the conformal Clifford algebras. We explain that every viewpoint change is given by a planar similarity and a perspective distortion encoded by the latitude angle of the camera. We model the perspective distortion by the point at infinity of the conformal model of the Euclidean plane described by D. Hestenesand we clarify the spinor representations of the similarities of the Euclidean plane. This leads us to represent the viewpoint changes by conformal transformations of ${\coprod }_{\alpha }{H}_{\alpha }$ for the Minkowski metric of the ambient space. Full article

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 intelligence should be present. We revisit a couple of elementary Euclidean geometry theorems discussed in the birth of Artificial Intelligence and a non-trivial inequality concerning triangles. GeoGebra succeeds in proving all these selected examples, while ChatGPT fails in one case. Our thesis is that both GeoGebra and ChatGPT could be used as complementary systems, where the natural language abilities of ChatGPT and the certified computer algebra methods in GeoGebra Discovery can cooperate in order to obtain sound and—more relevant—interesting results. Full article

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 with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback–Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics. Full article

While there are well-known synthetic methods in the literature for finding the image of a point under circular inversion in $l_2$-normed geometry (Euclidean geometry), there is no similar synthetic method in Minkowski geometry, also known as the geometry of finite-dimensional Banach spaces. In this study, we have succeeded in creating a synthetic construction of the circular inversion in $l_1$-normed spaces, which is one of the most fundamental examples of Minkowski geometry. Moreover, this synthetic construction has been given using the Euclidean circle, independently of the $l_1$-norm. Full article