Projective Geometry Research Articles

Harmonic curves and the beauty of projective geometry

AbstractThe purpose of this paper is to present projective geometry in a synthetic, visual and intuitive style through the central notion of harmonicity which leads to harmonic curves. This presentation includes new results, unpublished proofs of some classic theorems and a slight reformulation of its axiomatics.

Zero‐curvature subconformal structures and dispersionless integrability in dimension five

AbstractWe extend the recent paradigm “Integrability via Geometry” from dimensions 3 and 4 to higher dimensions, relating dispersionless integrability of partial differential equations to curvature constraints of the background geometry. We observe that in higher dimensions on any solution manifold, the symbol defines a vector distribution equipped with a subconformal structure, and the integrability imposes a certain compatibility between them. In dimension 5, we express dispersionless integrability via the vanishing of a certain curvature of this subconformal structure. We also obtain a “master equation” governing all second‐order dispersionless integrable equations in 5D, and count their functional dimension. It turns out that the obtained background geometry is parabolic of the type . We provide its Cartan‐theoretic description and compute the harmonic curvature components via the Kostant theorem. Then, we relate it to 3D projective and 4D conformal geometries via twistor theory, discuss symmetry reductions and nested Lax sequences, as well as give another interpretation of dispersionless integrability in 5D through Levi‐degenerate CR structures in 7D.

Covariant and manifestly projective invariant formulation of Thomas-Whitehead gravity

Thomas-Whitehead (TW) gravity is a recently formulated projectively invariant extension of Einstein-Hilbert gravity. Projective geometry was used long ago by Thomas to succinctly package equivalent paths encoded by the geodesic equation. Projective invariance in gravity has further origins in string theory through a geometric action constructed from the method of coadjoint orbits using the Virasoro algebra. A projectively invariant connection arises from this construction, a part of which is known as the diffeomorphism field. TW gravity exploits projective Gauss-Bonnet terms in the action functional to endow the diffeomorphism field with dynamics, while allowing the theory to collapse to general relativity in the limit that the diffeomorphism field vanishes and the connection becomes Levi-Civita. In the original formulation of TW gravity, the diffeomorphism field is projectively invariant but not tensorial and the connection is projectively invariant but not affine. In this paper we reformulate TW gravity in terms of projectively invariant tensor fields and a projectively invariant covariant derivative, derive field equations respecting these symmetries, and show that the field equations obtained are classically equivalent across formulations. This provides a “Rosetta Stone” between this newly constructed covariant and projective invariant formulation of TW gravity and the original formulation that was manifestly projective invariant, but not covariant. Published by the American Physical Society 2024

Four new families of NMDS codes with dimension 4 and their applications

For an [n,k,d]q linear code C, the singleton defect of C is defined by S(C)=n−k+1−d. When S(C)=S(C⊥)=1, the code C is called a near maximum distance separable (NMDS) code, where C⊥ is the dual code of C. NMDS codes have important applications in finite projective geometries, designs and secret sharing schemes. In this paper, we present four new constructions of infinite families of NMDS codes with dimension 4 and completely determine their weight enumerators. As an application, we also determine the locality of the dual codes of these NMDS codes and obtain four families of distance-optimal and dimension-optimal locally recoverable codes.

Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties

We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realization spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which an n-tuple of collinear points can be lifted to a nondegenerate realization of a point-line configuration. We show that forest configurations are liftable and characterize the realization space of liftable configurations as the solution set of certain linear systems of equations. Moreover, we study the Zariski closure of the realization spaces of liftable and quasi-liftable configurations, known as matroid varieties, and establish their irreducibility. Additionally, we compute an irreducible decomposition for their corresponding circuit varieties. Applying these liftability properties, we present a procedure to generate some of the defining equations of the associated matroid varieties. As corollaries, we provide a geometric representation for the defining equations of two specific examples: the quadrilateral set and the 3×4 grid. While the polynomials for the latter were previously computed using specialized algorithms tailored for this configuration, the geometric interpretation of these generators was missing. We compute a minimal generating set for the corresponding ideals.

Geometric correction of cone beam computed laminography based on projected address translation

Precise alignment of the system scan geometry is crucial to the reconstruction quality of computed laminography (CL). Different from computed tomography (CT), a challenging task for CL image reconstruction is to deal with rotational axes that are no longer symmetrical. In this paper, we take advantage of the fact that the difference between the inverse of the projected transverse coordinates of two points rotated about the axis of rotation under the CL scan geometry sums to zero when the projection angle differs by 180 deg. Based on this fact, the projection data under the scanned geometry with errors are aligned to the theoretical projection geometry. To validate this calibration method, a numerical printed circuit board (PCB) phantom was simulated. The results demonstrate that this method is capable of achieving high accuracy compared to the previous methods. The experimental results on printed circuit boards (PCB) demonstrate that the method can effectively improve geometric alignment accuracy of CL and obtain high-resolution reconstructed images.

Designs in finite classical polar spaces

AbstractCombinatorial designs have been studied for nearly 200 years. 50 years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the m-regular systems from projective geometry as the special case where the blocks are generators of the polar space. The first nontrivial such designs for $$t > 1$$ t > 1 were found by De Bruyn and Vanhove in 2012, and some more designs appeared recently in the PhD thesis of Lansdown. In this article, we investigate the theory of classical and subspace designs for applicability to designs in polar spaces, explicitly allowing arbitrary block dimensions. In this way, we obtain divisibility conditions on the parameters, derived and residual designs, intersection numbers and an analog of Fisher’s inequality. We classify the parameters of symmetric designs. Furthermore, we conduct a computer search to construct designs of strength $$t=2$$ t = 2 , resulting in designs for more than 140 previously unknown parameter sets in various classical polar spaces over $$\mathbb {F}_2$$ F 2 and $$\mathbb {F}_3$$ F 3 .

Eigenbasis for a weighted adjacency matrix associated with the projective geometry Bq(n)

In a recent article Projective geometries, Q-polynomial structures, and quantum groups Terwilliger (arXiv:2407.14964) defined a certain weighted adjacency matrix, depending on a free (positive real) parameter, associated with the projective geometry, and showed (among many other results) that it is diagonalizable, with the eigenvalues and their multiplicities explicitly written down, and that it satisfies the Q-polynomial property (with respect to the zero subspace).In this note we•Write down an explicit eigenbasis for this matrix.•Evaluate the adjacency matrix-eigenvector products, yielding a new proof for the eigenvalues and their multiplicities.•Evaluate the dual adjacency matrix-eigenvector products and directly show that the action of the dual adjacency matrix on the eigenspaces of the adjacency matrix is block-tridiagonal, yielding a new proof of the Q-polynomial property.

Coisotropic reduction in multisymplectic geometry

In this paper, we study coisotropic reduction in multisymplectic geometry. On the one hand, we give an interpretation of Hamiltonian multivector fields as Lagrangian submani- folds and prove that [Formula: see text]-coisotropic submanifolds induce a Lie subalgebra in the algebra of Hamiltonian [Formula: see text]-forms, similar to how coisotropic submanifolds in symplectic geometry induce a Lie subalgebra under the Poisson bracket. On the other hand, we extend the classical result of symplectic geometry of projection of Lagrangian submanifolds in coisotropic reduction to bundles of forms, which naturally carry a multisymplectic structure.

On parallelisms of PG(3,4) with automorphisms of order 2

Let PG(n,q) be the n-dimensional projective space over the finite field Fq. A spread in PG(n,q) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, and cryptography. All parallelisms of PG(3,2) and PG(3,3) are known. Parallelisms of PG(3,4) which are invariant under automorphisms of odd prime orders have also been classified. The present paper contributes to the classification of parallelisms in PG(3,4) with automorphisms of even order. We focus on cyclic groups of order four and the group of order two generated by a Baer involution. We examine invariants of the parallelisms such as the full automorphism group, the type of spreads and questions of duality. The results given in this paper show that the number of parallelisms of PG(3,4) is at least 8675365. Some future directions of research are outlined.

Computer Vision Algorithm Design in Image Processing Based on Projective Geometry

Image processing with computer vision, particularly in the realm of projective geometry, offers remarkable potential for various applications. Through the lens of projective geometry, images can be transformed, augmented, and reconstructed with precision, facilitating tasks such as image rectification, 3D reconstruction, and object tracking. Landmark estimation in computer vision is a vital task with broad applications across various domains. This process involves identifying key points or landmarks within images, enabling tasks such as facial recognition, object tracking, and gesture recognition. This paper, proposed a novel approach for landmark estimation in computer vision using Projective Geometry Landmark Estimation (PGLM). The proposed model aims to estimate the landmark features by a projective geometry model. With the estimation of the geometry features landmarks related to the facial, object, and medical images are computed. The PGLM model uses the point features for the location of the landmark features. In order to compare PGLM's performance to that of more conventional classification methods like Random Forest, K-Nearest Neighbors (KNN), and Support Vector Machine (SVM), simulation analysis is carried out. From what we can see, PGLM routinely beats these alternatives when we compare their accuracy, precision, recall, and F1 score. The findings stated the effectiveness of PGLM as a promising approach for landmark estimation in image processing tasks, paving the way for further advancements in this domain.

Using a Grassmann Graph to Recover the Underlying Projective Geometry

Using a Grassmann Graph to Recover the Underlying Projective Geometry

New quantum LDPC codes based on projective geometry

With the progress in techniques for correcting errors in quantum computing, quantum low density parity check (QLDPC) codes have gained increasing significance within the field of quantum error correction. This paper focuses on different strategies for the construction of QLDPC codes, which are based on all points and lines as well as partial points and lines from projective geometry. Finally, a series of simulation analyses are presented.

Phylogenetic Diversity Indices from an Affine and Projective Viewpoint

Phylogenetic diversity indices are commonly used to rank the elements in a collection of species or populations for conservation purposes. The derivation of these indices is typically based on some quantitative description of the evolutionary history of the species in question, which is often given in terms of a phylogenetic tree. Both rooted and unrooted phylogenetic trees can be employed, and there are close connections between the indices that are derived in these two different ways. In this paper, we introduce more general phylogenetic diversity indices that can be derived from collections of subsets (clusters) and collections of bipartitions (splits) of the given set of species. Such indices could be useful, for example, in case there is some uncertainty in the topology of the tree being used to derive a phylogenetic diversity index. As well as characterizing some of the indices that we introduce in terms of their special properties, we provide a link between cluster-based and split-based phylogenetic diversity indices that uses a discrete analogue of the classical link between affine and projective geometry. This provides a unified framework for many of the various phylogenetic diversity indices used in the literature based on rooted and unrooted phylogenetic trees, generalizations and new proofs for previous results concerning tree-based indices, and a way to define some new phylogenetic diversity indices that naturally arise as affine or projective variants of each other or as generalizations of tree-based indices.

Homogeneous Projective Coordinates for the Bondi–Metzner–Sachs Group

This paper studies the Bondi–Metzner–Sachs group in homogeneous projective coordinates because it is then possible to write all transformations of such a group in a manifestly linear way. The 2-sphere metric, the Bondi–Metzner–Sachs metric, asymptotic Killing vectors, generators of supertranslations as well as boosts and rotations of Minkowski spacetime are all re-expressed in homogeneous projective coordinates. Lastly, the integral curves of vector fields which generate supertranslations are evaluated in detail. This work paves the way for more advanced applications of the geometry of asymptotically flat spacetime in projective coordinates by virtue of the tools provided from complex analysis in several variables and projective geometry.

Architectural epistemology, existing buildings, and single-image modeling

This paper examines an alternative approach to creating a 3D digital model of an existing building on the basis of a single photograph. Rather than prioritizing comprehensive coverage or geometric accuracy, the method aims to highlight how the modelmaking process itself can generate architecturally specific knowledge. The paper describes modeling the Nishiki Market in Kyoto using principles of projective geometry and reverse perspective. By problematizing the construction process, the method discloses how a model may embody subjective interpretations and choices. The paper argues that this approach legitimizes producing models counter to prevailing conventions, as the process itself constitutes a form of situational architectural understanding, registered through traces of the modelmaker and the photograph’s perspective. Rather than foregrounding usefulness, this method values uncovering hidden assumptions and exposing the contingencies involved in constructing architectural knowledge.

Matroids Arising From Nested Sequences of Flats In Projective And Affine Geometries

Matroids Arising From Nested Sequences of Flats In Projective And Affine Geometries

Open-source data alternatives for regenerating urban road fine-grained features

The extraction of urban road features provides indispensable support to numerous high-accurate applications such as autonomous driving and urban high-definition mapping. However, approaches mainly focus on road connectivity, while often overlooking finer details of urban road constituent structures. Data that captures road details, such as LiDAR, may not be always readily available. This article proposes an operational framework for mapping fine-grained urban road features by integrating open-source data (OSD). The geometric measurement method is successively presented using projective geometry and prior knowledge for urban road sections. And a feature generation strategy is introduced to achieve and express the fine-grained road features. Compared with the corresponding large-scale topographic map (LTM) and available optical remote sensing image (AORSI), the proposed method regenerated fine-grained features of urban roads with m-level. It provides a cost-effective innovative and alternative method to acquiring fine-grained road datasets in other data-scarce regions.

Clinical Utility of Integrant’s FrameIt

This paper explores Integrant’s FrameIt, a novel external fixator based on the Taylor Spatial Frame (TSF). The TSF, invented by Dr. Charles Taylor, offers significant advantages over traditional Ilizarov frames in treating complex bone deformities and fractures. FrameIt leverages the TSF’s ability for precise six-axis adjustments and computer-assisted planning to achieve faster correction times and improved patient outcomes. The paper delves into the TSF’s background, highlighting its development as a solution to the limitations of the Ilizarov frame. It emphasizes the TSF’s reliance on principles from projective geometry, the Chasles’ Theorem and the Stewart Platform to achieve its superior accuracy. Furthermore, the paper explores the factors impacting the TSF’s success. It analyzes the influence of: Available and Emerging Technologies: The paper stresses the importance of utilizing advancements enhancing the TSF’s design and functionality Historical and Cultural Factors: The paper acknowledges the historical context that led to the development of external fixation systems, with the Ilizarov frame paving the way for the TSF Economic Factors: The paper discusses the economic considerations surrounding the TSF’s adoption, including equipment costs, procedures and aftercare. It highlights the limitations imposed by high costs on accessibility for some patients In conclusion, the paper positions Integrant’s FrameIt as a promising advancement in bone deformity correction. By building upon the TSF’s foundation and addressing economic barriers, FrameIt has the potential to revolutionize patient care in this field.

On the General Covariance of Natural Laws

This paper explores the concept of general covariance in natural laws through the lens of projective geometry and tensor algebra. By introducing the notions of covariance and contravariance using intuitive examples from projections and the scalar product, we illustrate how the covariance of natural laws ensures their universality and objectivity. We also discuss the role of symmetries and conservation principles in relation to the covariant nature of physical equations, highlighting the deep interplay between the mathematical structure of physical theories and the fundamental principles of nature.