Projective Geometry Research Articles

Geometry of the Mercator projection from 1569

Geometry of the Mercator projection from 1569

A Funk perspective on billiards, projective geometry and Mahler volume

A Funk perspective on billiards, projective geometry and Mahler volume

Boosting large‐scale construction project risk management: Application of the impact of building information modeling, knowledge management, and sustainable practices for optimal productivity

AbstractThis study investigates the relationship between Building Information Modeling (BIM) and knowledge management in large‐scale construction projects, emphasizing risk management, and increasing productivity. This study aims to identify the most important and impactful aspects of modern management techniques in the large‐scale construction sector to improve construction quality; focusing on cost reduction, energy conservation, and pollution reduction during construction. Employing a validated questionnaire from experts and the Analytical Hierarchy Process (AHP), the research prioritizes key factors, elucidating the pivotal role of BIM in understanding project geometry, efficient documentation, design flexibility, and reduced project delays. Ascertained through data analysis employing the pairwise comparisons method of the AHP analysis method, the compatibility of all comparisons and matrices arising from the decision‐making process of experts was below 0.1. The results establish BIM's technical capabilities, economic considerations, and pollution reduction as paramount, contributing significantly to improve large‐scale construction project risk management. The prioritization of the main criteria reveals their significance in the following order: BIM technical capabilities (0.281), BIM economic capabilities (0.231), pollution reduction (0.228), knowledge management capabilities (0.096), BIM time capabilities (0.090), and human resources management capabilities (0.074). BIM can improve risk management for large‐scale construction projects by: using the BIM model to understand the geometry and features of large‐scale construction projects; digitizing and organizing contract documents, project data, plans, work instructions, roles and responsibilities, cost, time, and energy analysis, and other project‐related documents for efficient project management; decreased duplication and project delays; reducing the need for on‐site personnel during implementation and CO2 emissions reduction. According to the results of the BIM model utilization to better understand the geometry and characteristics of large‐scale construction projects, the item was identified as the most significant and influential among the main components.

Implementation of analytic projective geometry for computer graphics

In their research, the authors actively exploit different branches of geometry. For geometric constructions, computer algebra approaches and systems are used. Currently, we are interested in computer geometry, more specifically, the implementation of computer graphics. The use of the projective space and homogeneous coordinates has actually become a standard in modern computer graphics. In other words, the problem is reduced to the application of analytic projective geometry. The authors failed to find a computer algebra system that could implement projective geometry in its entirety. Therefore, it was decided to partially implement computer algebra for visualization of algebraic relations. For this purpose, the Asymptote system was employed.

Implementation of Analytic Projective Geometry for Computer Graphics

Implementation of Analytic Projective Geometry for Computer Graphics

Adjoints of matroids

We first show that an adjoint of a loopless matroid is connected if and only if the original matroid is connected. By proving that the opposite lattice of a modular matroid is isomorphic to its extension lattice, we obtain that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice. This makes projective geometries become a key ingredient in characterizing the adjoint sequence ad0M,adM,ad2M,… of a connected matroid M. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to infinite projective geometries.

Integration-by-parts identities and differential equations for parametrised Feynman integrals

Integration-by-parts (IBP) identities and differential equations are the primary modern tools for the evaluation of high-order Feynman integrals. They are commonly derived and implemented in the momentum-space representation. We provide a different viewpoint on these important tools by working in Feynman-parameter space, and using its projective geometry. Our work is based upon little-known results pre-dating the modern era of loop calculations [16–19, 30, 31]: we adapt and generalise these results, deriving a very general expression for sets of IBP identities in parameter space, associated with a generic Feynman diagram, and valid to any loop order, relying on the characterisation of Feynman-parameter integrands as projective forms. We validate our method by deriving and solving systems of differential equations for several simple diagrams at one and two loops, providing a unified perspective on a number of existing results.

3-Uniform hypergraphs from vector spaces

The Fundamental Theorem of Projective Geometry states that, in a vector space, a permutation of vector lines preserving triples that span a vector plane is induced by a semi-linear automorphism. We consider a generalisation to triples of subspaces, not necessarily of the same dimension, spanning, or being contained in a subspace of fixed dimension. We determine all cases in which the permutation is necessarily induced by a semi-linear automorphism.

Projective geometry and spatial reasoning for STEM learning

Projective geometry is a prominent area in many fields including art, design, architecture, and mathematics, but how it can contribute to children’s spatial reasoning as well as a supplementary geometry to that of Euclid’s in school mathematics curricula raises the need for further consideration. With emphasis on the dynamic relationship between 2D and 3D objects, projective geometry not only offers important concepts but a meaningful context in which to develop spatial reasoning for STEM learning. In this article, we overview spatial reasoning as it relates to STEM and in particular, mathematics. Making a case for projective geometry, we present activities for the classroom, demonstrating how this mathematics topic connects to and opens new possibilities to promote spatial reasoning for STEM learning in the elementary grades.

Investigations of Solutions of Singularly Perturbed Differential Equations

Using the basic concept of projective geometry about the intersection of parallel lines, we have developed a new method for choosing integration paths. The integration paths must be decreasing from the starting point to the last point. This condition has remained. To check the correctness, consider a classic example. The advantages of the method are that the method is applicable when the eigenvalues of the matrix are real. The peculiarities of the real eigenvalues ​​of the matrix are that in this case the level lines degenerate at the point of stability change. As a result, the area under consideration is divided into several parts. Carrying out calculations along the selected integration paths, we obtain asymptotic estimates for solutions of singularly perturbed differential equations.

Projective Geometry as a Model for Hegel’s Logic

Recently, historians have discussed the relevance of the nineteenth-century mathematical discipline of projective geometry for early modern classical logic in relation to possible solutions to semantic problems facing it. In this paper, I consider Hegel’s Science of Logic as an attempt to provide a projective geometrical alternative to the implicit Euclidean underpinnings of Aristotle’s syllogistic logic. While this proceeds via Hegel’s acceptance of the role of the three means of Pythagorean music theory in Plato’s cosmology, the relevance of this can be separated from any fanciful “music of the spheres” approach by the fact that common mathematical structures underpin both music theory and projective geometry, as suggested in the name of projective geometry’s principal invariant, the “harmonic cross-ratio”. Here, I demonstrate this common structure in terms of the phenomenon of “inverse foreshortening”. As with recent suggestions concerning the relevance of projective geometry for logic, Hegel’s modifications of Aristotle respond to semantic problems of his logic.

A Matroid-Based Automatic Prover and Coq Proof Generator for Projective Incidence Geometry

A Matroid-Based Automatic Prover and Coq Proof Generator for Projective Incidence Geometry

Estimation of planar angles from non-orthogonal imaging.

Photogrammetry-based methods are traditionally used to estimate the geometrical parameters using optical images. These methods employ specific equipment, computationally sophisticated and expensive algorithms, and utilize projective geometry to reconstruct real-life scenes up to a scale. In this work, we used a computationally less-expensive method for sparse reconstruction to estimate the planar angles using two-view geometry and linear algorithms from non-orthogonal images acquired by a smartphone camera. First, intrinsic camera parameters were determined. Next, scale-invariant feature transform was used to identify the correspondence points from each pair of images. Epipolar constraint was applied on all these points to determine the essential matrix using the eight-point algorithm. Thereafter, extrinsic camera parameters were estimated from the essential matrix and combined with the intrinsic matrix to get the camera projection matrix. Finally, linear triangulation was used to get the sparse point cloud representing the scene. Planar angles were estimated by backprojecting the chosen image points and applying simple vector algebra on the obtained 3D points. The method was successful in estimating the planar angles in less than 10s on non-curved edges with an average error of 3% by using only ten images. Given the simplicity of methods used, this technique can be integrated into a smartphone for on-site measurements as well as large deformations.

BIM-Based Building Performance Simulation Analysis: A Multi-Parameter-Driven Approach to Building Energy Efficiency and Carbon Reduction

Global warming and other environmental problems are increasing the demand for high-performance buildings. The development of high-performance computers and building information models has a significant impact on buildings’ energy simulation. High-performance building design needs to comprehensively consider geography, climate, material, cost and other factors which is a highly complex multidisciplinary research problem. Therefore, it is urgent to use advanced modeling and simulation technology to realize it, involving Building Information Modelling (BIM), parametric design and cloud platform. Comprehensive simulation of building performance refers to a multidisciplinary collaborative design, and the correlation between research objects and parameters should be achieved by complex programming design. This study integrates BIM, computer, cloud computing and other technologies to simulate BIM-based building energy consumption performance. Based on project information, geometry information and physical properties exhibited by materials stored in BIM model, the energy analysis model is created. Revit–Dynamo API functions are employed to generate a novel BIM model in Revit after automatically changing and transferring user-defined parameters. BIM energy consumption model is converted into Green Building eXtended Markup Language (GBXML) file and uploaded to Green Building Studio (GBS) cloud server. The optimal project solution is yielded by retrieving the energy consumption simulation results of BIM models with a range of parameters. The case study shows that building volume, glass material, window-wall ratio and window height have significant influence on energy consumption targets of buildings. In hot-summer and cold-winter areas, the total energy consumption of glass materials with high insulation and reflection coefficient is small. The window size slightly impacts the annual lighting energy consumption, but it has significant influence on the annual air conditioning energy consumption, with a maximum increase of about 22%. Finally, the application advantages and limitations of the framework in high-performance building design and its application prospects in energy-saving building design are discussed.

Constructing a straight line intersecting four lines

In this paper, we determine the set S of straight lines L0 that have intersections with four given distinct lines L1,…,L4 in R3. If any two of the four given lines are skew, i.e., not co-planar, Bielinski and Lapinska used techniques in projective geometry to show that there are either zero, one, or two elements in the set S. Using linear algebra techniques, we determine S and show that there are no, one, two or infinitely many elements L0 in S, where the last case was overlooked in the earlier paper. For the sake of completeness, we provide a comprehensive determination of all the elements L0 in S if at least two of the four given lines are co-planar. In this scenario, there may also be zero, one, two, or infinitely many solutions.

Fostering Ethnomathematics in Projective Geometry: A Hypnoteaching Framework for Effective Distance Learning

This study evaluated the impact of integrating students' cultural backgrounds into online learning to improve understanding and participation. It employed "Simple House Theory" and projective geometry in designing traditional houses to make complex mathematical concepts relatable. The Ethnomathematics project and the Hypnoteaching model were two instructional strategies that successfully enhanced active student participation in distance learning.Data was collected from a joint course between Universitas Negeri Makassar (UNM Makassar), Indonesia, and Tunghai University, Taiwan. The study involved 36 students: 20 from UNM Makassar’s international mathematics education program (Group I) and 16 from Tunghai University’s applied mathematics department (Group II). Various tools like surveys, tests, and projects were employed for data collection. The survey had 23 Likert-scale questions, allowing a thorough evaluation of student feedback. Quantitative analysis of this data revealed that incorporating the Ethnomathematics projects and Hypnoteaching model into the curriculum significantly increased student engagement in remote learning. This was further evidenced by improved student performance in project work. The research underscores the benefits of culturally relevant teaching methods in enhancing engagement and understanding in online mathematics education. It highlights the effectiveness of embedding local cultural elements into online learning environments, creating more engaging and relatable experiences. The study suggests the potential of these approaches in other educational contexts and disciplines for future research.

On the Photon Vertex and Half-Integer Spin

When rewriting the photon vertex of quantum electrodynamics in terms of geometrical quantities, various elements can be mapped directly to objects and properties known from classical projective geometry (PG). Elements of P 5 when mapped to line reps in P 3 exhibit their intrinsic Lorentz invariance associated to automorphisms of the Plücker-Klein quadric , and line reps when expressed by point or plane coordinates introduce (one-parameter) pencils, or formally gl(2,ℝ), or gl(1,ℍ) which covers su(2)⊕u(1). This introduces binary forms and, using a potential approach of central forces, Schrödinger or Laplace equations and the respective special functions, as well as the projective generation of quadrics like in Dirac’s approach which legitimates Clifford algebra elements as linear factors in invariant theory and the quadratic algebra to represent geometry. Physically, this identification allows for the classical concept of moments in terms of tetrahedrons which on the one hand relates to previous work on SU(4) and SU*(4) in quantum representations. On the other hand, it relates to the classical physical definitions, however, exhibiting a factor 2 between contemporary (euclidean) moments and the tetrahedral construction used in the vertex. Finally, we discuss the equilibrium conditions with respect to gauge and Yang-Mills theories in general as well as the related objects and their transformation theory.

Pilot study on high-resolution radiological methods for the analysis of cerebrospinal fluid (CSF) shunt valves

ObjectivesDespite their life-saving capabilities, cerebrospinal fluid (CSF) shunts exhibit high failure rates, with a large fraction of failures attributed to the regulating valve. Due to a lack of methods for the detailed analysis of valve malfunctions, failure mechanisms are not well understood, and valves often have to be surgically explanted on the mere suspicion of malfunction.The presented pilot study aims to demonstrate radiological methods for comprehensive analysis of CSF shunt valves, considering both the potential for failure analysis in design optimization, and for future clinical in-vivo application to reduce the number of required shunt revision surgeries. The proposed method could also be utilized to develop and support in situ repair methods (e.g. by lysis or ultrasound) of malfunctioning CSF shunt valves. Materials and methodsThe primary methods described are contrast-enhanced radiographic time series of CSF shunt valves, taken in a favorable projection geometry at low radiation dose, and the machine-learning-based diagnosis of CSF shunt valve obstructions. Complimentarily, we investigate CT-based methods capable of providing accurate ground truth for the training of such diagnostic tools. Using simulated test and training data, the performance of the machine-learning diagnostics in identifying and localizing obstructions within a shunt valve is evaluated regarding per-pixel sensitivity and specificity, the Dice similarity coefficient, and the false positive rate in the case of obstruction free test samples. ResultsContrast enhanced subtraction radiography allows high-resolution, time-resolved, low-dose analysis of fluid transport in CSF shunt valves. Complementarily, photon-counting micro-CT allows to investigate valve obstruction mechanisms in detail, and to generate valid ground truth for machine learning-based diagnostics.Machine-learning-based detection of valve obstructions in simulated radiographies shows promising results, with a per-pixel sensitivity >70%, per-pixel specificity >90%, a median Dice coefficient >0.8 and <10% false positives at a detection threshold of 0.5. ConclusionsThis ex-vivo study demonstrates obstruction detection in cerebro-spinal fluid shunt valves, combining radiological methods with machine learning under conditions compatible to future in-vivo application.Results indicate that high-resolution contrast-enhanced subtraction radiography, possibly including time-series data, combined with machine-learning image analysis, has the potential to strongly improve the diagnostics of CSF shunt valve failures. The presented method is in principle suitable for in-vivo application, considering both measurement geometry and radiological dose. Further research is needed to validate these results on real-world data and to refine the employed methods.In combination, the presented methods enable comprehensive analysis of valve failure mechanisms, paving the way for improved product development and clinical diagnostics of CSF shunt valves.

Near-MDS codes from maximal arcs in PG(2,q)

The singleton defect of an [n,k,d] linear code C is defined as s(C)=n−k+1−d. Codes with S(C)=0 are called maximum distance separable (MDS) codes, and codes with S(C)=S(C⊥)=1 are called near maximum distance separable (NMDS) codes. Both MDS codes and NMDS codes have good representations in finite projective geometry. MDS codes over Fq with length n and n-arcs in PG(k−1,q) are equivalent objects. When k=3, NMDS codes of length n are equivalent to (n,3)-arcs in PG(2,q). In this paper, we deal with the NMDS codes with dimension 3. By adding some suitable projective points in maximal arcs of PG(2,q), we can obtain two classes of (q+5,3)-arcs (or equivalently [q+5,3,q+2] NMDS codes) for any prime power q. We also determine the exact weight distribution and the locality of such NMDS codes and their duals. It turns out that the resultant NMDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes.

Pole Estimation and Optical Navigation Using Circle of Latitude Projections

Images of both rotating celestial bodies (e.g., asteroids) and spheroidal planets with banded atmospheres (e.g., Jupiter) can contain features that are well-modeled as a circle of latitude (CoL). At large distances, the projections of these CoLs appear as ellipses in images collected by cameras or telescopes onboard exploration spacecraft. This work shows how CoL projections may be used to determine the pole orientation and covariance for a spinning asteroid. In the case of a known planet modeled as an oblate spheroid, it is shown how similar CoL projections may be used for spacecraft localization. These methods are developed using the principles of projective geometry. Numerical results are provided for simulated images of asteroid Bennu (for pole orientation) and of Jupiter (for spacecraft localization).