Projective Plane Research Articles

Advanced Planar Projection Contour (PPC): A Novel Algorithm for Local Feature Description in Point Clouds.

Local feature description of point clouds is essential in 3D computer vision. However, many local feature descriptors for point clouds struggle with inadequate robustness, excessive dimensionality, and poor computational efficiency. To address these issues, we propose a novel descriptor based on Planar Projection Contours, characterized by convex packet contour information. We construct the Local Reference Frame (LRF) through covariance analysis of the query point and its neighboring points. Neighboring points are projected onto three orthogonal planes defined by the LRF. These projection points on the planes are fitted into convex hull contours and encoded as local features. These planar features are then concatenated to create the Planar Projection Contour (PPC) descriptor. We evaluated the performance of the PPC descriptor against classical descriptors using the B3R, UWAOR, and Kinect datasets. Experimental results demonstrate that the PPC descriptor achieves an accuracy exceeding 80% across all recall levels, even under high-noise and point density variation conditions, underscoring its effectiveness and robustness.

Connection of Conic and Cylindrical Map Projections

In previous papers that have dealt with cylindrical map projections as limiting cases of conical projections, standard or equidistant parallels were used in the derivations. This paper shows that this is not necessary and that it is sufficient to use parallels that preserve length. In addition, unlike other approaches, in this article the limiting cases of conic projections are derived in the most natural way, by deriving the equations of cylindrical projections from the equations of conic projections in a rectangular system in the projection plane using a mathematical concept of limits. It is shown that such an approach is possible, but not always, so it should be used carefully, or even better, avoided in teaching and studying map projections.

Three Conics Determine a Cubic

Given a cubic K in the real projective plane. Then for each point P there is a conic CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{{P}}$$\\end{document} associated to P. The conic CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{{P}}$$\\end{document} is called the polar conic of K with respect to the pole P. We investigate the situation when three conics C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, and C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} are polar conics of K with respect to the poles P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_1$$\\end{document}, P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_2$$\\end{document}, and P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_3$$\\end{document}, respectively. In particular, we give an elementary proof—without using any results from algebraic geometry—that any three conics C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} in general position, satisfying only a non-degeneracy condition, determine a unique cubic K and three points P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_1$$\\end{document}, P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_2$$\\end{document}, P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_3$$\\end{document}, such that C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} are polar conics of K with respect to the three poles P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_1$$\\end{document}, P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_2$$\\end{document}, P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_3$$\\end{document}. This can be seen as a higher degree variant of von Staudt’s Theorem.

UAV building point cloud contour extraction based on the feature recognition of adjacent points distribution

Building point cloud contour is widely used in urban planning and building identification. The existing method cannot work well in the extraction of UAV building point cloud contour points because the low point cloud accuracy of UAV, and effect of noise. In this paper we propose extraction method of UAV building point cloud contour based on the adjacent points distribution. Firstly, we construct an information entropy model for determining optimal neighborhood points, and then use the eigenvectors corresponding to the maximum and minimum eigenvalues obtained by PCA to construct initial projection plane Π1 and Π2. Secondly, we propose the fine-tuning model of the projection plane Π2 by using the aggregation characteristics of neighborhood points, and construct the extraction model of fold points with three parameter constraints. Finally, we propose the information entropy model of points probabilities in different quadrants on the plane Π1 to extract boundary points. Real UAV point cloud data is used to test the performance and parameters of the proposed method, experiment results show that the proposed method is superior to OPAHT, 2D line detection and region clustering segmentation methods in the performance of UAV building point cloud contour extraction. The proposed method can accurately extract the point cloud contour points of UAV buildings.

Algebraic properties of binomial edge ideals of Levi graphs associated with curve arrangements

In this article, we study algebraic properties of binomial edge ideals of Levi graphs associated with certain plane curve arrangements. Using combinatorial properties of Levi graphs, we discuss the Cohen-Macaulayness of binomial edge ideals of Levi graphs associated to some curve arrangements in the complex projective plane, like the d-arrangement of curves and the conic-line arrangements. We also discuss the existence of certain induced cycles in the Levi graphs of these arrangements and obtain lower bounds for the regularity of powers of the corresponding binomial edge ideals.

Symmetric projection optimizer: concise and efficient solving engineering problems using the fundamental wave of the Fourier series

The fitness function value is a kind of important information in the search process, which can be more targeted according to the guidance of the fitness function value. Most existing meta-heuristic algorithms only use the fitness function value as an indicator to compare the current variables as good or bad but do not use the fitness function value in the search process. To address this problem, the mathematical idea of the fitting is introduced into the meta-heuristic algorithm, and a symmetric projection optimizer (SPO) is proposed to solve numerical optimization and engineering problems more efficiently. The SPO algorithm mainly utilizes a new search mechanism, the symmetric projection search (SP) method. The SP method quickly completes the fitting of the projection plane, which is located through the symmetry of the two points and finds the minima in the projection plane according to the fitting result. Fitting by using the fitness function values allows the SP to find regions where extreme values may exist more quickly. Based on the SP method, exploration and exploitation strategies are constructed, respectively. The exploration strategy is used to find better regions, and the exploitation strategy is used to optimize the discovered regions continuously. The timing of the use of the two strategies is designed so that the SPO algorithm can converge faster while avoiding falling into local optima. The effectiveness of the SPO algorithm is extensively evaluated using seven test suites, including CEC2017, CEC2019, CEC2020, and CEC2022. It is also compared with two sets of 19 recent competitive algorithms. Statistical analyses are performed using five metrics such as the Wilcoxon test, the Friedman test, and variance. Finally, the practicality of the SPO algorithm is verified by four typical engineering problems and a real spacecraft trajectory optimization problem. The results show that the SPO algorithm can find superior results in 94.6% of the comparison tests and is a promising alternative for solving real-world problems.

Enumeration of Sets of Mutually Orthogonal Latin Rectangles

We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR homogeneous.In the course of doing this, we perform a complete enumeration of non-isotopic sets of $t$ mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$. Specifically, we keep track of homogeneous sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR transitive.We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are homogeneous and/or transitive in each step of the construction process. We use the prefix stepwise to refer to sets of MOLR with this property.Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular graphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.

The ordinal of dynamical degrees of birational maps of the projective plane

The ordinal of dynamical degrees of birational maps of the projective plane

A minimal and non-alternative realisation of the Cayley plane

The compact 16-dimensional Moufang plane, also known as the Cayley plane, has traditionally been defined through the lens of octonionic geometry. In this study, we present a novel approach, demonstrating that the Cayley plane can be defined in an equally clean, straightforward and more economic way using two different division and composition algebras: the paraoctonions and the Okubo algebra. The result is quite surprising since paraoctonions and Okubo algebra possess a weaker algebraic structure than the octonions, since they are non-alternative and do not satisfy the Moufang identities. Intriguingly, the real Okubo algebra has SU3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SU}\\left( 3\\right) $$\\end{document} as automorphism group, which is a classical Lie group, while octonions and paraoctonions have an exceptional Lie group of type G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {G}_{2}$$\\end{document}. This is remarkable, given that the projective plane defined over the real Okubo algebra is nevertheless isomorphic and isometric to the octonionic projective plane which is at the very heart of the geometric realisations of all types of exceptional Lie groups. Despite its historical ties with octonionic geometry, our research underscores the real Okubo algebra as the weakest algebraic structure allowing the definition of the compact 16-dimensional Moufang plane.

Investigation of the functional and biomechanical effect of instrument-assisted soft tissue mobilization technique in individuals with asymptomatic dynamic knee valgus - Randomized controlled trial

Although there are studies showing that myofascial release will increase muscle force production, the contribution of its application alone to muscle force production has not been examined. Aim of the study is to investigate the effect of instrument-assisted soft tissue mobilization (IASTM) on eccentric strength, frontal plane projection angle (FPPA), dynamic (DPS), and static postural stability (SPS), femoral internal rotation (FIR) angle in females with dynamic knee valgus (DKV).A total of 44 recreationally active females with asymptomatic DKV (age: 21,39 ± 1,79, body mass index: 20,09 ± 2,45) participated and were randomly assigned to either control group (CG) or IASTM group (IASTMG).Participants' eccentric contraction strength, FPPA, DPS, SPS, and FIR on the involved leg were measured pre- and post. IASTM application was applied to IASTMG for 6 weeks, twice a week, for 5 min, using Graston Technique® instruments on gluteus medius. CG received no intervention. In comparison of ECS difference values, change in IASTMG was found to be statistically significantly higher than CG (p = .004; p < .01). There was no statistical difference in comparison of FIR and FPPA values (respectively p = .213, p = .360; p < .05). In SPS and DPS evaluation, a statistically significant improvement was observed in favor of IASTMG in comparison of both intergroup and difference values (p < .05 for all). Strength gain without exercise can increase postural stability, but it isn't sufficient to correct faulty movement patterns. We recommend adding IASTM to injury prevention programs, but there is a need to investigate the effect of IASTM with technique correction feedback.

Changes in dynamic quadriceps angle in patients with chronic ankle instability

Background: Changes in dynamic Quadriceps Angle (Q angle) had been studied in healthy subjects and individuals with other musculoskeletal disorders; however, in chronic ankle instability (CAI), it has not yet been determined. The purpose of this study was to investigate the difference in dynamic Q angle of patients with unilateral chronic ankle instability and those without chronic ankle instability. Methods: One-hundred and four participants with and without chronic ankle instability from both genders aged between 18-30 years, with a BMI range of 17.9-27.3, were enrolled into 2 equal groups; a study group and a control group, with mean values of age of the study and control groups (20.8±2.4) and (20.9±3) years respectively and of BMI (24.1±4.3) and (23.8±2.9) kg/m2 respectively. Dynamic Q angle measurement is carried out using dynamic knee valgus (DKV) through the frontal plane projection angle (FPPA) during single limb squat. Results: A statistically significant difference in FPPA angle was observed between the affected limb of the CAI group and controls (P=0.001). Conclusion: The present study suggested that unilateral CAI display an increased knee valgus, suggesting an additional proximal kinematic alteration and chronic maladaptation in the CAI population.

On the existence of Spot It! decks that are not projective planes

Abstract The game Spot It! is played with a deck of cards in which every pair of cards has exactly one matching symbol and the aim is to be the fastest at finding the match. It is known that finite projective planes correspond to decks in which every card contains n symbols and every symbol appears on n cards. In this paper we relax the hypothesis on the number of cards on which a symbol appears: we study symmetric decks in which every symbol appears the same number of times and we introduce the concept of maximal deck, providing a sufficient condition for a deck to have this property. We also produce various examples of interesting decks which do not correspond to projective planes.

Model-based monocular 6-degree-of-freedom pose tracking for asteroid

In this paper, we present a novel vision-based framework to track the 6-DoF pose of an asteroid in real time with the 3D contour of the asteroid as a feature. During pose tracking, at the beginning time of tracking, the tracking system is initialized by a pose retrieval method. At each subsequent time instant, given the 3D mesh model of an asteroid, with the initial pose and its covariance given by the square root cubature Kalman Filter (SCKF), the 3D mesh segments constituting the 3D asteroid contour are efficiently extracted from the 3D mesh model. Then, in the input asteroid image, we search the image points corresponding to the extracted 3D segments within the searching range defined by the initial pose and its covariance. After that, the asteroid pose is determined in real time by minimizing the angles between the back-projection lines of the searched image points and the projection planes of the corresponding 3D segments, which is much more robust to the position change of the asteroid and asteroid size. The covariance matrix of the pose is inferred from the Cartesian noise model in the first order. Eventually, the SCKF is derived from the second-order auto regression to generate the final pose estimate and give the initial pose and its covariance for the next time instant. The synthetic trials quantitatively validate the real-time performance, robustness, and accuracy of our algorithm in dark space, different imaging distances, lighting conditions, image noise, model error, and initial pose error, and meanwhile, the real trial qualitatively shows the effectiveness of our method.

Enumerative geometry of del Pezzo surfaces

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin [Duke Math. J. 170 (2021), pp. 1291–1375]. We also include some explicit calculations for the projective plane, which confirm some folklore conjectures in this case.

Team–Work: The Olympics 1925 and 1931

For the cultural history of industrialized nations, particularly in the economies of the Global North, the period between 1890 and 1930 is associated with modernisms, as successive cultural movements that were formally innovative, highly subjective, yet also self-reflexive of their institutional and social functions. These movements proclaimed themselves as avant-garde; as cultural vanguards that visualize, materialize, and sound out abstract ideas in new artistic forms and practices. Many modernisms, from Futurism to social realism, regarded the human body as a performative projection plane for expansive ideas about movement and mobility, often conflating social reform with physical freedom, and mass action with political agency.

[formula omitted]-cohomology projective planes from Enriques surfaces in odd characteristic

We give a complete classification of Qℓ-cohomology projective planes with isolated ADE-singularities and numerically trivial canonical bundle in odd characteristic. This leads to a beautiful relation with certain Enriques surfaces which parallels the situation in characteristic zero, yet displays intriguing subtleties.

Self-Calibrating Gaze Estimation With Optical Axes Projection for Head-Mounted Eye Tracking

Gaze estimation suffers from burdensome personal calibration or complex all-device calibration. Self-calibrating methods can meet this challenge but depend on scenes and sacrifice accuracy. We propose a flexible and accurate gaze estimation approach calibrated implicitly with potential gaze patterns. By constructing an optical axis projection (OAP) plane and a visual axis projection (VAP) plane simultaneously, the optical axis and the visual axis can be represented as 2-D points, i.e., the OAP and VAP, which have a similarity transformation, indicating the linear consistency of OAP patterns with gaze patterns. Hence, a 3-D gaze estimation model using the OAP as an eye feature to predict the VAP is built. The unknown parameters are calculated separately by linearly aligning OAP patterns to natural and easily detectable gaze patterns. Experimental results show that the proposed gaze estimation approach is more accurate than state-of-the-art head-mounted gaze estimation methods, which require explicit calibration or multi-scene saliency.

Almost optimal measurement scheduling of molecular Hamiltonian via finite projective plane

We propose an efficient and almost optimal scheme for measuring molecular Hamiltonians in quantum chemistry on quantum computers, which requires 2N2 distinct measurements in the leading order with N being the number of molecular orbitals. It achieves the state-of-the-art by improving a previous proposal by Bonet-Monroig [], which exhibits 103N2 scaling in the leading order. We develop a method based on a finite projective plane to construct sets of simultaneously measurable operators contained in molecular Hamiltonians. Each measurement only requires a depth-O(N) circuit consisting of O(N2) one- and two-qubit gates under the Jordan-Wigner and parity mapping, assuming the linear connectivity of qubits on quantum hardwares. We perform numerical simulation of our method for molecular Hamiltonians of hydrogen chains. We count the number of sets of simultaneously measurable operators generated by our method and estimate the number of measurement shots required to achieve the small standard deviation of the energy expectation value for precise quantum chemistry calculations. Because evaluating expectation values of molecular Hamiltonians is one of the major bottlenecks in the applications of quantum devices to quantum chemistry, our finding is expected to accelerate such applications. Published by the American Physical Society 2024

Parallels in Cartography: Standard, Equidistantly Mapped and True Length Parallels

In the literature on map projections, we regularly encounter the name standard parallel or standard parallels. However, it is obvious that a unique definition of a standard parallel is not universally accepted. To fully clarify the meaning of standard parallels, the author proposes the notion of equidistantly mapped parallels, which has not been common in the literature so far. Equidistantly mapped parallels can be in the direction of the parallel or in the direction of the meridian. Here, it is shown that every standard parallel is also an equidistantly mapped parallel, but that the reverse need not be true. If the parallel is mapped equidistantly in the direction of the parallel, then its length in the projection plane is equal to the length of that parallel on the sphere. The opposite does not have to be true, i.e., if the length of the image of the parallel in the projection plane is equal to the length of the parallel on the sphere, this does not mean that the parallel was mapped equidistantly. In addition to standard and equidistant parallels, the concept of parallels of true length also appears in the theory of map projections. They should also be distinguished from standard and equidistant parallels.

Anisotropic elastic and thermal properties and damage tolerance of CrH: A first-principles calculation

The elastic properties, thermal properties, and damage tolerance of one cubic (C-CrH) and two hexagonal (H1-CrH and H2-CrH) CrH hydrides were investigated using first-principles calculation based on density functional theory (DFT). The results showed that C-CrH is brittle, while H1-CrH and H2-CrH are ductile. Compared to the hexagonal CrH, C-CrH has the highest hardness, the largest damage tolerance, and the largest fracture toughness values. Three-dimensional (3D) surface configurations and two-dimensional (2D) planar projections were used to estimate the elastic anisotropy, and the elastic anisotropy is in the order of H2-CrH > C-CrH > H1-CrH. In addition, C-CrH has the highest minimum thermal conductivity among these CrH hydrides. CrH hydrides are anisotropic in minimum thermal conductivity with the order of H1-CrH > C-CrH > H2-CrH.