Projective Plane Research Articles

Poncelet’s theorem for conics in any position and any characteristic

Poncelet’s theorem states that if there exists an n-sided polygon which is inscribed in a given conic C and circumscribed about another conic D, then there are infinitely many such n-gons. Proofs of this theorem that we are aware of, including Poncelet’s original proof and the celebrated modern proof by Griffiths and Harris, assume the two conics to be in general position (that is, not tangent or at least not osculating), or be defined over , or both. Here we show that Poncelet’s theorem holds for any two conics C and D in the projective plane over an algebraically closed field k of any characteristic other than two. If C and D are osculating and , our result shows that there always exist infinitely many polygons of sides that are inscribed in C and circumscribed about D. We also describe the situation in characteristic two in the appendix.

Singularities of Holomorphic Codimension One Foliations of the Complex Projective Plane

We prove that any holomorphic codimension 1 foliation on the complex projective plane has at most one singularity up to the action of an ad-hoc birational map. Consequently, any algebraic foliation on the affine plane has no singularities up to the action of a suitable birational self map of the complex projective plane into itself.

Disjoint sets in projective planes of small order

In this paper, results of a computer search for disjoint sets associated with maximal arcs and unitals in projective planes of order 16, and disjoint sets associated with unitals in projective planes of orders 9 and 25 are reported. It is shown that the number of pairs of disjoint unitals in planes of order 9 is exactly four, and new pairs and triples of disjoint degree 4 maximal arcs are shown to exist in some of the planes of order 16. New bounds on the number of 104-sets of type (4, 8) and 156-sets of type (8, 12) are achieved. A combinatorial method for finding new maximal arcs, new unitals, and new v-sets of type (m, n) is introduced. All disjoint sets found in this study are explicitly listed.

Testing the impact of Remotely Operated Vehicle (ROVs) camera angle on community metrics of temperate mesophotic organisms: A 3D model-based approach

While Remotely Operated Vehicle (ROV) technologies have rapidly increased our understanding of deeper water mesophotic ecosystems, there are some inherent problems in their use when estimating the abundance of sessile benthic organisms. Specifically, for rocky and coral reefs there has been recent debate about the importance of the angle of the ROV camera to the substrate, which is difficult to standardise, and how variation in this angle might impact abundance estimates and therefore our ability to detect change in mesophotic ecosystems. Here we simulated a series of 3D rocky temperate mesophotic communities based on real data with three different levels of habitat complexity (high, medium, and low). We ‘sampled’ these communities in a virtual space varying the angle of the camera to the substrate, to assess the influence of camera angle on percent cover estimates and overall benthic community composition. We found that increases in the ROV to substrate angle overestimated the percent cover (per degree of camera angle) of gorgonians (large: R2 = 0.16; small: R2 = 0.18), tubular sponges (R2 = 0.11), and ascidians (R2 = 0.05–0.13), and underestimated the abundance of Crustose Coraline Algae (R2 = 0.1, likely due to shading) in the most complex habitats. While we also found some significant differences in the overall benthic community and data dispersion for the more complex habitats, angle explained only a small amount of total variation (<6%) and significant dispersion was only found between 0- and 50-degree angles in high and medium complexity habitats. We conclude that while ROV to surface angle should be a consideration when sampling deep water reefs, its influence on standard measurements that we estimate with photography is likely to be low, and restricted to errors in the abundance of organisms with a high ratio of height to planar projection area (e.g., gorgonians).

Vertex points are not enough: Monocular 3D object detection via intra- and inter-plane constraints

Existed methods for 3D object detection in monocular images focus mainly on the class of rigid bodies like cars, while more challenging detection like the cyclist is less studied. Therefore, we propose a novel 3D monocular object detection method to improve the accuracy of detection objects with large differences in deformation by introducing the geometric constraints of the object 3D bounding box plane. Considering the map relationship of projection plane and the keypoint, we firstly introduce the geometric constraints of the object 3D bounding box plane, adding the intra-plane constraint while regressing the position and offset of the keypoint itself, so that the position and offset error of the keypoint are always within the error range of the projection plane. For the inter-plane geometry relationship of the 3D bounding box, the prior knowledge is incorporated to optimize the keypoint regression allowing for improved the accuracy of depth location prediction. Experimental results show that the proposed method outperforms some other state-of-the-art methods on cyclist class, and obtains competitive results in the field of real-time monocular detection.

Unitals in Projective Planes of Order 25

Unitals in Projective Planes of Order 25

Examination of overhead and single-leg squat kinematics and lower extremity injury in female collegiate athletes.

Overhead squat (OHS) and single-leg squat (SLS) tests have been used as screening tools to identify injury risk in athletic populations. Yet, no study has examined the association between specific trunk and lower extremity (LE) kinematic measures acquired during OHS and SLS performance and LE injury incidence in female collegiate athletes. The purpose of this study was to examine the association between LE and trunk kinematics captured during OHS and SLS performance and LE injury in female collegiate athletes. This was a prospective cohort study. One hundred eight Division I female collegiate athletes underwent OHS and SLS testing before the start of their competitive season. A Microsoft Kinect sensor using Athletic Movement Assessment software (PhysiMax®) was used to capture kinematic variables: hip flexion, knee flexion, knee frontal plane projection angle (FPPA), and trunk flexion. Participants were tracked during one competitive season for incidence of LE injury. Logistic regression models were used to examine the association between OHS and SLS measures and injury. Twenty-three (21.3%) participants suffered a LE injury. No significant associations were found between any OHS (odds ratio [OR] range: 0.91-1.08) or SLS (OR range: 0.81-1.22) kinematic measure and incidence of LE injury. Sagittal plane LE and trunk kinematics and knee FPPA during OHS and SLS performance were not associated with LE injury in this cohort of athletes. Our findings do not support the use of these select kinematic measures captured during OHS and SLS testing as stand-alone injury risk assessments in female collegiate athletes.

The Apparent Structure of Dense Sidon Sets

The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from projective planes through a similar construction. We classify the Sidon sets arising in this manner from desarguesian planes and find essentially no new examples. There are many further examples arising from nondesarguesian planes.&#x0D; We conjecture that all dense Sidon sets arise from finite projective planes in this way. If true, this implies that all abelian groups of most orders do not have dense Sidon subsets. In particular if $\sigma_n$ denotes the size of the largest Sidon subset of $\mathbb{Z}/n\mathbb{Z}$, this implies $\liminf_{n \to \infty} \sigma_n / n^{1/2} &lt; 1$.&#x0D; We also give a brief bestiary of somewhat smaller Sidon sets with a variety of algebraic origins, and for some of them provide an overarching pattern.

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.

Low-energy points on the sphere and the real projective plane

We present a generalization of a family of points on S2, the Diamond ensemble, containing collections of N points on S2 with very small logarithmic energy for all N∈N. We extend this construction to the real projective plane RP2 and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.

Topological structure of optimal flows on the Girl's surface

We investigate the topological structure of flows on the Girl's surface which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of self-intersection. First, we describe the cellular structure of the Boy's and Girl's surfaces and prove that there are unique images of the project plane in the form of a $2$-disk, in which the opposite points of the boundary are identified and this boundary belongs to the preimage of the $1$-skeleton of the surface. Second, we describe three structures of flows with one fixed point and no separatrices on the Girl's surface and prove that there are no other such flows. Third, we prove that Morse-Smale flows and they alone are structurally stable on the Boy's and Girl's surfaces. Fourth, we find all possible structures of optimal Morse-Smale flows on the Girl's surface. Fifth, we obtain a classification of Morse-Smale flows on the projective plane immersed on the Girl's surface. And finally, we describe the isotopic classes of these flows.

Topological approaches to knotted electric charge distributions

Consider a knot K in S^3 with uniformly distributed electric charge. Whilst solutions to the Laplace equation in terms of Dirichlet integrals are readily available, it is still of theoretical and physical interest to understand the qualitative behavior of the potential, particularly with respect to critical points and equipotential surfaces. In this paper, we demonstrate how techniques from geometric topology can yield novel insights from the perspective of electrostatics. Specifically, we show that when the knot is sufficiently close to a planar projection, we get a lower bound on the size of the critical set based on the projection’s crossings, improving a 2021 result of the author. We then classify the equipotential surfaces of a charged knot distribution by tracking how the topology of the knot complement restricts the Morse surgeries associated to the critical points of the potential.

Comparison of Single-Camera-Based Depth Estimation Technology for Digital Twin Model Synchronization of Underground Utility Tunnels

Digital twin technology can be used for disaster safety management of underground utility tunnels, which are social infrastructures. To verify its efficiency in an underground utility tunnel, it must be implemented and monitored in the real world. Therefore, the state of information about an object and the position of an object in the real world, which can change upon movement, should be correspondingly reflected in the digital model. Underground utility tunnels are monitored through closed circuit television (CCTV) cameras because the main infrastructure facilities are installed in a long and narrow form, which limits access to them. Therefore, synchronization of a digital twin model of an underground utility tunnel requires a method for identifying the location of an object through an installed CCTV camera. This study compares the DenseDepth transfer learning method and the coordinate system conversion method through floor plan projection to estimate depth using a single CCTV camera installed in an underground utility tunnel. The coordinate system conversion method through floor plane projection showed an error of 0.23 m to 1.5 m within a 50 m range. The DenseDepth transfer learning method showed an error of 1 m to 3 m; therefore, it was considered unsuitable for distance estimation. The coordinate system conversion method through floor plane projection showed a smaller error even at a longer distance than the DenseDepth transfer learning method. Therefore, the coordinate system conversion method through floor plane projection was considered more suitable than the DenseDepth transfer learning method for depth estimation using a single camera.

Dynamic attitude measurement taken by monocular line scan vision.

Attitude measurements are important parameters for condition monitoring of precision machines. We develop an angle measurement method based on monocular line scan vision to dynamically measure the attitude of shafts. The proposed measurement system, which consists of a single line scan camera, a parallel laser light source, and a triangular mirror, does not require fixation of a sensor on the shaft. The measurement system uses the line scan camera to capture the projection of the shaft. The light source is diffused by a cylindrical lens to yield parallel directional light. Then the parallel directional light is reflected by a fixed-angle mirror and projected onto the projection plane. The captured image can be considered a discrete time sequence. The attitude measurement principle based on linear monocular vision is derived. A double light-spot centroid algorithm is developed to obtain the centroid coordinates of the projection points. The proposed method is investigated by a simulation and experiment. The proposed method can also monitor the vibration of the shaft. The experimental results prove that the proposed method is effective and highly accurate.

On Automorphic Forms of Small Weight for Fake Projective Planes

On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists, but not unique, otherwise). Earlier we conjectured that any such cubic root must be acyclic. In the present note we give two short proofs of this statement and show acyclicity of some other line bundles on the fake projective planes with at least $9$ automorphisms. Similarly to our earlier work we employ simple representation theory for non-abelian finite groups. The first proof is based on the observation that if some line bundle is non-linearizable with respect to a finite abelian group, then it should be linearized by a finite, \emph{non-abelian}, Heisenberg group. For the second proof, we also demonstrate vanishing of odd Betti numbers for a class of abelian covers, and use linearization of an auxiliary line bundle as well.

Differential geometry of complex projective plane conics

Differential geometry of complex projective plane conics

The shadow of supertranslated black hole

The supertranslated black hole proposed by Hawking, Perry, and Strominger might provide a resolution to the information paradox, which is usually defined by a complicated space-time metric. In this paper, we figure out the shadow for the supertranslated black hole by making use of supertranslated 4-velocities and the trajectories of the light rays. Based on this approach, although the photon sphere gets distorted and the position of the shadow on the projection plane is shifted by the supertranslation vector due to the supertranslation hairs, the size and shape of the shadow remain the same as those of bald black hole. However, the shift of the position of shadow should be understood by the choice of coordinate and then we conclude that there are no distinguishable effects for the supertranslated black hole.

Beyond the Standard Model with Six-Dimensional Spinors

Six-dimensional spinors with Spin(3,3) symmetry are utilized to efficiently encode three generations of matter. E8(−24) is shown to contain physically relevant subgroups with representations for GUT groups, spacetime symmetries, three generations of the standard model fermions, and Higgs bosons. Pati–Salam, SU(5), and Spin(10) grand unified theories are found when a single generation is isolated. For spacetime symmetries, Spin(4,2) may be used for conformal symmetry, AdS5→dS4, or simply broken to Spin(3,1) of a Minkowski space. Another class of representations finds Spin(2,2) and can give AdS3 with various GUTs. An action for three generations of fermions in the Majorana–Weyl spinor 128 of Spin(4,12) is found with Spin(3) flavor symmetry inside E8(−24). The 128 of Spin(12,4) can be regarded as the tangent space to a particular pseudo-Riemannian form of the octo-octonionic Rosenfeld projective plane E8(−24)/Spin(12,4)=(OsxO)P2.

Trinomials, torus knots and chains

Letn&gt;mn&gt;mbe fixed positive coprime integers. Forv&gt;0v&gt;0, we give a topological description of the setΛ(v)\Lambda (v), consisting of points[x:y:z][x:y:z]in the complex projective plane for which the equationxζn+yζm+z=0x\zeta ^n +y \zeta ^m+z=0has a root with normvv. It is shown that the setΩ(v)=PC2∖Λ(v)\Omega (v)= {\mathbb P_{\mathbb C}} ^2 \setminus \Lambda (v)hasn+1n+1components. Moreover, the topological type of each component is given. The same results hold forΛ\LambdaandΩ=PC2∖Λ\Omega ={\mathbb P_{\mathbb C}}^2 \setminus \Lambda, whereΛ\Lambdadenotes the set obtained as the union of all the complex tangent lines to the33-sphere at the points of the torus knot, that is, the knot obtained by intersecting{[x:y:1]∈PC2:|x|2+|y|2=1}\{[x:y:1] \in \mathbb {P}_{\mathbb C}^2 : |x|^2+|y|^2=1\}and the complex curve{[x:y:1]∈PC2:ym=xn}\{[x:y:1] \in {\mathbb P_{\mathbb C}} ^2 : y^m=x^n\}. Finally, we use the linking number of a distinguished family of circles and the torus knot to give a numerical invariant which determines the components ofΩ\Omegain a unique way.

Comparison of angular cephalometric measurements between CBCT mid-sagittal plane projections and conventional lateral cephalometric radiographs

The purpose of this study was to verify the angular measurement values of conventional Lateral Cephalometric Radiographs (LCRs) and Cone-Beam Computed Tomography (CBCT) Mid- Sagittal Plane (MSP) projections when using the same cephalometric analysis method.