Geometric Regularity Research Articles

Universal Language of Geometry: Geometrical Grid – The Nature of Space

The space of an artistic plane, as well as architectural space, has certain regularities, which different theories explain in various ways. Geometric regularities in space are visible through the theoretical and practical work of artists and architects. For centuries, ever since Vitruvius, theorists have been looking for valid criteria to analyze the beauty of architecture. Iosif Brodsky goes as far as to say that beauty is the purpose of evolution, while other authors define geometry as the source of all beauty. Based on these hypotheses, in the 20th century the words of the architect Buckminster Fuller must be highlighted. He says that geometry is the essence of everything that surrounds us. The importance of geometry in space conception and its analysis are stated throughout this paper. Article received: March 29, 2018; Article accepted: April 10, 2018; Published online: September 15, 2018; Scholarly analysis or debateHow to cite this article: Petrović, Milica. "Universal Language of Geometry: Geometrical Grid – The Nature of Space." AM Journal of Art and Media Studies 16 (2018): 69−84. doi: 10.25038/am.v0i16.255

Reconstruction of Three-Dimensional (3D) Indoor Interiors with Multiple Stories via Comprehensive Segmentation

The fast and stable reconstruction of building interiors from scanned point clouds has recently attracted considerable research interest. However, reconstructing long corridors and connected areas across multiple floors has emerged as a substantial challenge. This paper presents a comprehensive segmentation method for reconstructing a three-dimensional (3D) indoor structure with multiple stories. With this method, the over-segmentation that usually occurs in the reconstruction of long corridors in a complex indoor environment is overcome by morphologically eroding the floor space to segment rooms and by overlapping the segmented room-space with partitioned cells via extracted wall lines. Such segmentation ensures both the integrity of the room-space partitions and the geometric regularity of the rooms. For spaces across floors in a multistory building, a peak-nadir-peak strategy in the distribution of points along the z-axis is proposed in order to extract connected areas across multiple floors. A series of experimental tests while using seven real-world 3D scans and eight synthetic models of indoor environments show the effectiveness and feasibility of the proposed method.

Tracking and validation techniques for topographically organized tractography

Topographic regularity of axonal connections is commonly understood as the preservation of spatial relationships between nearby neurons and is a fundamental structural property of the brain. In particular the retinotopic mapping of the visual pathway can even be quantitatively computed. Inspired from this previously untapped anatomical knowledge, we propose a novel tractography method that preserves both topographic and geometric regularity. We make use of parameterized curves with Frenet-Serret frame and introduce a highly flexible mechanism for controlling geometric regularity. At the same time, we incorporate a novel local data support term in order to account for topographic organization. Unifying geometry with topographic regularity, we develop a Bayesian framework for generating highly organized streamlines that accurately follow neuroanatomy. We additionally propose two novel validation techniques to quantify topographic regularity. In our experiments, we studied the results of our approach with respect to connectivity, reproducibility and topographic regularity aspects. We present both qualitative and quantitative comparisons of our technique against three algorithms from MRtrix3. We show that our method successfully generates highly organized fiber tracks while capturing bundle anatomy that are geometrically challenging for other approaches.

Space Subdivision in Indoor Mobile Laser Scanning Point Clouds Based on Scanline Analysis.

Indoor space subdivision is an important aspect of scene analysis that provides essential information for many applications, such as indoor navigation and evacuation route planning. Until now, most proposed scene understanding algorithms have been based on whole point clouds, which has led to complicated operations, high computational loads and low processing speed. This paper presents novel methods to efficiently extract the location of openings (e.g., doors and windows) and to subdivide space by analyzing scanlines. An opening detection method is demonstrated that analyses the local geometric regularity in scanlines to refine the extracted opening. Moreover, a space subdivision method based on the extracted openings and the scanning system trajectory is described. Finally, the opening detection and space subdivision results are saved as point cloud labels which will be used for further investigations. The method has been tested on a real dataset collected by ZEB-REVO. The experimental results validate the completeness and correctness of the proposed method for different indoor environment and scanning paths.

Asymptotically locally Euclidean/Kaluza–Klein stationary vacuum black holes in five dimensions

We produce new examples, both explicit and analytical, of bi-axisymmetric stationary vacuum black holes in five dimensions. A novel feature of these solutions is that they are asymptotically locally Euclidean, in which spatial cross-sections at infinity have lens space |$L(p,q)$| topology, or asymptotically Kaluza–Klein so that spatial cross-sections at infinity are topologically |$S^1\times S^2$|⁠. These are nondegenerate black holes of cohomogeneity 2, with any number of horizon components, where the horizon cross-section topology is any one of the three admissible types: |$S^3$|⁠, |$S^1\times S^2$|⁠, or |$L(p,q)$|⁠. Uniqueness of these solutions is also established. Our method is to solve the relevant harmonic map problem with prescribed singularities, having target symmetric space |$SL(3,\mathbb{R})/SO(3)$|⁠. In addition, we analyze the possibility of conical singularities and find a large family for which geometric regularity is guaranteed.

Effect of process parameters on the quality of aluminium alloy Al5Si deposits in wire and arc additive manufacturing using a cold metal transfer process

ABSTRACTA 3D print device using a cold metal transfer arc welding station to melt a metallic filler wire is developed to build aluminium part by optimising the process parameters. First tests achieved using standard pre-recorded process parameters allow to study the effect of the travel speed and the average welding power on the geometrical characteristics of mono-layer deposits and on walls built by layers superposition. Finally, a parametric study of the effect of each process parameter controlling the shape of the arc current or voltage and the filler wire feeding is carried out in order to try to improve the geometrical regularity of the deposits, and to better understand the effect of each parameter on the melting of the filler wire, its transfer on the support plate, and the geometry of the formed bead.

The Geometrical Regularization for Covering Irregular Bases with Karbandi

Karbandi is an arched covering pattern in Persian architecture which provides the interior of a structure with a particular geometric elegance and integrates architecture and structure in the creation of a meaningful spaces. The use of these structures is highly demanded in contemporary Persian architecture but requires a creative, modern design and construction strategy in order to meet new requirements, since traditional methods for implementing them causes them to be applicable only on regular bases. Here the authors investigate a solution to broaden their use over irregular geometry. This paper also aims at determining the possibility of administrating Karbandi geometry in the boundaries of an irregular polygonal shape. The result of this paper shows that it is possible to calculate and implement a Karbandi geometry in irregular bases when some specific conditions are met.

Regularity for degenerate evolution equations with strong absorption

In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of p-Laplacian type (2≤p<∞) under a strong absorption condition:Δpu−∂u∂t=λ0u+qinΩT:=Ω×(0,T), where 0≤q<1. This model is interesting because it yields the formation of dead-core sets, i.e., regions where non-negative solutions vanish identically. We shall prove sharp and improved parabolic Cα regularity estimates along the set F0(u,ΩT)=∂{u>0}∩ΩT (the free boundary), where α=pp−1−q≥1+1p−1. Some weak geometric and measure theoretical properties as non-degeneracy, positive density, porosity and finite speed of propagation are proved. As an application, we prove a Liouville-type result for entire solutions. A specific analysis for Blow-up type solutions will be done as well. The results are new even for dead-core problems driven by the heat operator.

The Other Battle of Algiers

The Other Battle of Algiers

Urban Planning by Le Corbusier According to Praxeological Knowledge

The city is formed as a mosaic of various elements which affect its attractiveness. These elements range from location attributes, through economic opportunities, to social aspects. Therefore, urbanity and urban planning should be considered in a multi-dimensional context. In the paper we address the problem of urban planning by Le Corbusier according to praxeological and system knowledge. From praxeological point of view an active human being takes his/her choice between various possibilities by preferring one of these possibilities to the others, and by manifesting it by her actions. The same applies to the design process. Due to this fact, the scientific design process can be treated as a systematic rational reconstruction of the designer’s behaviour. Such a reconstruction requires previous reflection on designer’s work, as well as some consideration and design experience, thus know-how knowledge based on methodological knowledge. In the paper several city visions of Le Corbusier, as well as the characteristics and organisation of his design process are analysed. Le Corbusier’s innovative design ideas resulted from industrialisation changes and motorisation accelerating progress, which gave foundation to a new urban array. This array based on strict geometric forms, regularity and repetition determining standard. Thanks to his theories, Le Corbusier established principles of modern city construction and planning. Although some doubts were expressed as to the scale of centralisation of the cities designed by him and his class-based conception, he was awarded that overall welfare of the individual living in a city was the quality of built environment. Therefore, his designed creations were not only functional but they also produced emotions. The analysis of his prolific design activities allows us to state that the organisation of his architectural and urban planning process was very efficient and complex. The city concepts proposed by him were the subject of analysis by generations of designers. Also now, they can still be the basis for modelling virtual and navigable cities by modern planners. Le Corbusier’s comprehensive approach to modern city planning showed that research activities, that is theoretical thinking, and production activities, that is practice, are linked methodically. Therefore, urban planning should be understood not only as a projection of the possibilities of architecture, but as a multidisciplinary process. Due to this fact, an urban plan, as a result of that process, should be a synthesis of various social, industrial and economic aspects.

Tetrahedral and spherical representations of the periodic system

The s, p, d and f blocks of the elements, as delimited by Charles Janet in 1928, can be represented as parallel slices of a regular tetrahedron. They also fit neatly on to the surface of a sphere. The reasons for this are discussed and the possible objections examined. An attempt is made to see whether there are philosophical implications of this unexpected geometrical regularity. A new tetrahedral design in transparent plastic is presented.

The Spatial Identity of Historic Cairo, Part one: Typology, Geometry, and Geographic Distribution

It is fundamental to maintain the local places’ identity, character and trajectory of evolution in order to create a sustainable process of urbanization.This study introduces a quantitative inventory of urban spaces in Cairo, in the early 19th century, extracted from historic maps and indexes. Urban spaces are classified in terms of geometric typology (linear and central), geometric regularity of their footprints, sizes and geographical distribution.The extracted information highlighted the quantitative domination of linear spaces and of irregularity in space footprint. It also showed a noticeable insignificant value of central spaces. However, the insignificant value of central spaces presented spatial identity of power demonstration in historic Cairo; irregularity and linearity presented sociocultural and economic identities.

Strong Stationarity Conditions for Optimal Control of Hybrid Systems

We present necessary and sufficient optimality conditions for finite-time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush–Kuhn–Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine (PWA) system can be written as the optimizer of a mathematical program, which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with PWA dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation.

Design and volume optimization of space structures

We study the design and optimization of statically sound and materially efficient space structures constructed by connected beams. We propose a systematic computational framework for the design of space structures that incorporates static soundness, approximation of reference surfaces, boundary alignment, and geometric regularity. To tackle this challenging problem, we first jointly optimize node positions and connectivity through a nonlinear continuous optimization algorithm. Next, with fixed nodes and connectivity, we formulate the assignment of beam cross sections as a mixed-integer programming problem with a bilinear objective function and quadratic constraints. We solve this problem with a novel and practical alternating direction method based on linear programming relaxation. The capability and efficiency of the algorithms and the computational framework are validated by a variety of examples and comparisons.

Directional geometric histogram feature extraction and applications

Image feature has been a hot research topic within the field of computer vision, with a wide scope of direct impacts on detection, recognition, image retrieval and pose estimation, etc. In this paper, we propose a novel image feature: Directional Geometric Histogram (DGH) which adopts directional geometric approximation from the geometric Bandelet transform to enhance the description distinctiveness and selectivity among monocular images, particularly by renovating the histogram of geometric regularity to characterize local image context with human objects. Other than the image geometry defined over edges, our approach can well preserve inner and outer patterns of contours with strict geometry. We have compared the proposed method with classic global features and conducted comprehensive experiments in human detection, pose estimation as well as scene recognition tasks on various datasets. Final evaluation results show that the dimensionality of the DGH feature can be reduced to less than half of the original size, which is also sparse while keeping competitive discriminatory effectiveness and distinctiveness in such visual tasks. Besides its relaxed computational requirement and off-the-shelf theoretical backup, the method is in the meanwhile quite promising for potential fields in video surveillance, pattern identification, etc.

Geometric Regularity, Symmetry and the Perceived Beauty of Simple Shapes

Geometric regularity is introduced as a simplicity metric for understanding the esthetics of basic polygons. In Experiment 1, undergraduates rated the perceived beauty of triangles varying in geometric regularity defined by their side-length standard deviation. Each type of triangle was used: equilateral, isosceles, right, and scalene. Ratings increased with an increase in regularity and were highest for symmetric shapes. The results were replicated in a second experiment using each type of quadrilateral: square, rectangle, parallelogram, rhombus, kite, isosceles, and trapezoid. Although ratings in Experiment 2 also increased with regularity, symmetry did not perfectly predict ranked order of preference. Symmetry, partial symmetry, and familiarity are discussed as factors that can account for the results. On the basis of the shapes used in this study, we conclude that simplicity in the form of geometric regularity is usually but not always preferred.

Local regularity and compactness for the p-harmonic map heat flows

Abstract We study a geometric analysis and local regularity for the evolution of p {{p}} -harmonic maps, called p {{p}} -harmonic map heat flows. Our main result is to establish a criterion for a uniform local regularity estimate for regular p {{p}} -harmonic map heat flows, devising some new monotonicity-type formulas of a local scaled energy. The regularity criterion obtained is almost optimal, comparing with that of the corresponding stationary case. As application we show a compactness of regular p {{p}} -harmonic map heat flows with energy bound.

Synthetic aperture radar image compression based on multi-scale geometric transforms

Image representation in separable orthogonal basis cannot take advantage of geometrical regularity contained in basic images. When, explored efficiently geometrical regularity improves image compression. In this paper, we propose to experiment and compare an adaptive multi-scale geometric decomposition for synthetic aperture radar (SAR) image compression, called multi-scale bandelet transform, and a non-adaptive multi-scale geometric representation called ridgelet transform. The second generation of bandelet transform adopted in this work, is constructed in discrete domain with bandeletisation of warped wavelet transform along the optimal direction of geometric flow that minimises the Lagrangian. We discuss the criteria and results to assess SAR image compression performances using wavelet, bandelet, and ridgelet transforms. Our experiments revealed that during the compression phase, the speckle noise is removed from the SAR images inducing further improvements of the coding efficiency. In order, to evaluate the robustness of bandelet transform, we have proposed a progressive compression scheme based on the second generation of bandelet transform combined to SPIHT encoder, which is generally integrated with the wavelet transform.

Volume Reconstruction from Slices

We propose a variational approach for the reconstruction of a volume from slices. The reconstructed set is obtained as a minimizer of a geometric regularity criterion, either the perimeter or the Willmore energy, with inclusion-exclusion constraints associated with the cross sections. We propose a phase field approximation of this model, and we analyze it when the regularity criterion is the perimeter. We derive simple and accurate numerical schemes for both the perimeter-based and the Willmore-based formulations, and we illustrate with several numerical examples the performances of our approach, which proves to be effective for a large category of constraints.

On the geometric regularity conditions for the 3D Navier–Stokes equations

We prove geometrically improved version of Prodi–Serrin type blow-up criterion. Let v and ω be the velocity and the vorticity of solutions to the 3D Navier–Stokes equations and denote {f}+=max{f,0}, QT=R3×(0,T). If {(v×ω∣ω∣)⋅Λβv∣Λβv∣}+∈Lx,tγ,α(QT) with 3/γ+2/α≤1 for some γ>3 and 1≤β≤2, then the local smooth solution v of the Navier–Stokes equations on (0,T) can be continued to (0,T+δ) for some δ>0. We also prove localized version of a special case of this. Let v be a suitable weak solution to the Navier–Stokes equations in a space–time domain containing z0=(x0,t0), let Qz0,r=Bx0,r×(t0−r2,t0) be a parabolic cylinder in the domain. We show that if either {(v×ω∣ω∣)⋅∇×ω∣∇×ω∣}+∈Lx,tγ,α(Qz0,r) with 3γ+2α≤1, or {(v∣v∣×ω)⋅∇×ω∣∇×ω∣}+∈Lx,tγ,α(Qz0,r) with 3γ+2α≤2, (γ≥2, α≥2), then z0 is a regular point for v. This improves previous local regularity criteria for the suitable weak solutions.