Arbitrary Coordinate System Research Articles

A Virtual Reality Tool for Accuracy Assessment of 3D Models in an Immersive Virtual Environment

Abstract. Accurate validation and assessment techniques are essential for ensuring the reliability of spatial reconstructions derived from photogrammetry, enabling well-informed decision-making across diverse domains. This study presents a Virtual Reality (VR) based accuracy assessment tool tailored for evaluating the accuracy and quality of 3D models generated by Unmanned Aerial Vehicles (UAVs). Leveraging the Unity game engine platform, our workflow entails three key steps: aligning real-world coordinates with an arbitrary Unity coordinate system, transforming the positions of Ground Control Points (GCPs) from field survey to the arbitrary system using a reference GCP, and marking observed points on the 3D models. Absolute and Relative Root Mean Square Errors (RMSE), Mean Errors (ME), and Standard Deviation of errors (SD) are computed within the virtual environment via the game object transform properties. The error distributions around each GCP are visually depicted using Unity game engine components for enhanced interaction and comprehension. The efficacy of the tool is validated through experimentation on four 3D models generated from varying camera angles during UAV data capture. The tool provides the opportunity to directly interact with the 3D models and visualize the errors, which is quite distinct from traditional methods. Using the developed tool, results were obtained to indicate that configurations employing camera angles of 60° + 75º exhibit notable performance in terms of relative and absolute accuracy.

Aspect ratio-dependent volume resistivity in unidirectional composites: Insights from electrical conduction behavior

The electrical properties of carbon fibers serve as the foundation for the multifunctional applications of carbon fiber-reinforced composite structures. In scenarios that exploit the electrical characteristics of materials, accurate estimation of electrical resistivity stands as a critical factor. This study endeavors to elucidate the electrical conduction behaviors in unidirectional composites with different fiber orientation angles (0°, 15°, 30°, 45°, 60°, 75°, and 90°) and aspect ratios, thereby deriving the volume resistivity within an arbitrary Cartesian coordinate system. Employing thermal infrared imaging technology and finite element analysis, we identified distinctive electrical conduction behaviors associated with aspect ratios in carbon fiber composite plates. Notably, a critical aspect ratio exists wherein the diagonal yarn is the only conductive path between two electrodes. Below this critical threshold, no direct conductive path exists, and current flows through the shortest distance between parallel yarns. Conversely, beyond the critical aspect ratio value, multiple yarns form conductive paths between the two electrodes. Based on the electrical conduction behavior of unidirectional composites under different angles and aspect ratios, the volume resistivity with finite boundaries was derived and examined under an arbitrary Cartesian coordinate basis.

Propagation attenuation of elastic waves in multi-row infinitely periodic pile barriers: A closed-form analytical solution

Periodically arranged media exhibit attenuation zones for elastic waves, and the related periodic structures play a crucial role in the design of seismic isolation and vibration mitigation. From the perspective of wave theory, this paper presents a closed-form analytical solution for the propagation attenuation of elastic waves through infinitely periodic, multi-row pile barriers. The innovative approach begins by deriving the extended Graf’s addition theorem to transform wave potentials across arbitrary coordinate systems, overcoming the constraints of traditional formulas that limit pile centers to linear configurations. Subsequently, it completes the wave function expansion by utilizing the phase differences in the frequency domain of wave fields around different periodic elements, thus skillfully integrating the effects of incident and scattered wave fields. For the first time, this study delivers exact expressions for the scattering of various types of plane waves (SH, SV, and P) by multi-row pile barriers with an infinite periodic array, addressing the complexities of large pile numbers in previous theoretical studies. The proposed solution increases computational efficiency by tens of times and markedly reduces the majority of resource usage, improving the analysis of complex vibration isolation systems involving numerous piles. Building on the accuracy verification and convergence analysis, several numerical examples are conducted to explore the effects of pile row, pile spacing, and pile arrangement on the propagation attenuation. The findings elucidate the mechanisms driving vibration reduction design using periodic wave barrier, furthering its application in engineering structures.

A recursive trapezoid-based algorithm designed to compute the strength- and stiffness-related geometric properties of beams with polygonal cross-sections

A new recursive trapezoid-based algorithm is proposed for calculating the strength- and stiffness-related geometric characteristics of polygonal beam sections. By taking as input the coordinate vertices of the polygon, the approach may effectively be applied to determine the cross-section area and the position of its centroid along with the axial area moments of inertia and product of inertia relative to the cross-sections’ centroidal axes. As the method combines the theory of planar moments with the well-known trapezoid numerical method, it provides a straightforward tabular procedure that can be easily implemented using any spreadsheet software. In addition, aiming to determine the principal area moments of inertia, a computation matrix formalism based on the theory of orthogonal diagonalization of symmetric matrices is put forward as an alternative to the classical method of vanishing the first derivative and solving for the principal directions. Some benchmark examples are worked out to prove the relevance and accuracy of the proposed approach for several beam sections ranging from the most common triangular and quadrilateral ones to very complex open and closed polygonal profiles. The solution presented here appears to be virtually foolproof compared to the conventional one, as it does not necessitate the decomposition of complex polygonal cross-section area into elementary surfaces of different shapes, then use the parallel axes theorem, and so forth; rather, it requires only insertion of the coordinates of the vertices of the polygon, turn by turn in the clockwise order, relative to an arbitrary cartesian coordinate system, and no further steps.

КОМП'ЮТЕРНЕ МОДЕЛЮВАННЯ НЕСТАЦІОНАРНОГО ПРОЦЕСУ РОЗВИТКУ НАДЗВУКОВОГО ПЕРЕРОЗШИРЕНОГО СТРУМЕНЮ

Questions of numerical simulation of non-stationary process of the development of a supersonic overexpanded jet are considered. Unsteady Reynolds-averaged Navier-Stokes equations (URANS) of compressible flows, written in an arbitrary coordinate system, along with the Spalart-Allmaras one-equation differential turbulence model are applied. The calculations of non-stationary formation of an air cold supersonic jet during the blowing through a conical Laval nozzle (simulation of engine start-up) were made. The physical features of the shock and acoustic waves generation and propagation are discussed. In the stationary phase an overexpanded free supersonic turbulent jet is formed, which includes a Mach disc in its structure. The obtained results were compared with experimental data and calculations by other authors based on the ANSYS Fluent and FloEFD CFD tools.

Boyer–Lindquist Space-Times and Beyond: Metamaterial Analogues for Arbitrary Space-Times

Analogue space-times (and in particular metamaterial analogue space-times) have a long varied and rather complex history. Much of the previous related work to this field has focused on spherically symmetric models; however, axial symmetry is much more relevant for mimicking astrophysically interesting systems that are typically subject to rotation. Now it is well known that physically reasonable stationary axisymmetric space-times can, under very mild technical conditions, be put into Boyer–Lindquist form. Unfortunately, a metric presented in Boyer–Lindquist form is not well adapted to the “quasi-Cartesian” metamaterial analysis that we developed in our previous articles on “bespoke analogue space-times”. In the current article, we shall first focus specifically on various space-time metrics presented in Boyer–Lindquist form, and subsequently determine a suitable set of equivalent metamaterial susceptibility tensors in a laboratory setting. We shall then turn to analyzing generic space-times, not even necessarily stationary, again determining a suitable set of equivalent metamaterial susceptibility tensors. Perhaps surprisingly, we find that the well-known ADM formalism proves to be not particularly useful, and that it is instead the dual “threaded” (Kaluza–Klein–inspired) formalism that provides much more tractable results. While the background laboratory metric is (for mathematical simplicity and physical plausibility) always taken to be Riemann flat, we will allow for arbitrary curvilinear coordinate systems on the flat background space-time. Finally, for completeness, we shall reconsider spherically symmetric space-times, but now in general spherical polar coordinates rather than quasi-Cartesian coordinates. In summary, this article provides a set of general-purpose calculational tools that can readily be adapted for mimicking various interesting (curved) space-times by using nontrivial susceptibility tensors in general (background-flat) laboratory settings.

Обобщённое неравенство Крамера – Рао в случае совместного пеленгования по азимуту и углу места в условиях сложной электромагнитной обстановки

In this work, a generalized expression is obtained for the Cramer–Rao lower bound in the case of a joint estimation of azimuth and elevation angle in an arbitrary coordinate system in which the vector complex radiation pattern is determined, under conditions of a complex electromagnetic environment, including the presence of noise correlation in spatial channels and differences in its intensity. A theorem is formulated for the Fisher information of joint estimates of azimuth and elevation angle in the case of a multi-channel detector-direction finder with an antenna system described in the selected coordinate system by a vector complex radiation pattern. Dependences of the root-mean-square error of direction finding in azimuth and elevation on the signal-to-noise ratio and true directions to the radio source are obtained, calculated using known and generalized expressions for the Cramer–Rao lower bound. The condition for the applicability of the existing and proposed in the article expression for the Fisher information matrix is obtained. The concept of the “energy” center of the antenna system is given. To confirm the reliability of the obtained generalized expressions for the lower bound of the dispersion of the Cramer–Rao inequality, statistical modeling is carried out. The use of the obtained expressions in practice will allow the most optimal synthesis of antenna devices depending on the structure and characteristics of the direction finder antenna system, the number of antennas and the characteristics of the radio receiving path. Generalized expressions for the lower Cramer–Rao bound are the foundation for organizing the operation of intelligent radio monitoring systems, allowing for a well-founded choice of the optimal antenna array structure that provides the required direction-finding efficiency.

Anisotropic media with singular slowness surfaces

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

Automatic Fingerprint Data Labeling Using WiFi Signal and Smartphone Camera for Indoor Positioning

WiFi fingerprinting has been one of the most practical approaches for implementing an indoor positioning system. However, the need to measure location labels for fingerprint data has hindered the deployment of WiFi fingerprint‐based positioning systems. To reduce the cost of location labeling, several methods for automatically labeling locations or bypassing location labeling have been proposed. However, these methods still need to measure location labels for partial fingerprint data, or the estimated location labels are inaccurate. In this paper, we propose Heimdall, an automatic fingerprint data labeling method using a smartphone camera and WiFi signal for an indoor positioning system. Since Heimdall utilizes camera geometry for accurate automatic labeling, it aims to find the location labels of unlabeled fingerprint data composed of WiFi received signal strengths (RSSs) and photos collected using smartphones in multiple reference points (RPs). Using structure from motion (SfM) with all photos in the unlabeled fingerprint dataset, the location and orientation of the cameras can be recovered on an arbitrary world coordinate system. However, the world coordinate system derived by SfM is warped. In other words, it is different from the real‐world coordinate system where we express the location labels since SfM has similarity ambiguity, including scale, rotation, and translation ambiguity. Heimdall finds the location labels by estimating and applying an unwarping transformation that removes those ambiguities. We use two types of geometric constraints for estimating the unwarping transformation. First, several geometric entities in the warped‐world coordinates and real‐world coordinates are extracted from the photos and prior information about the indoor layout. The unwarping transformation transforms the geometric entities in warped coordinates to those in real‐world coordinates. Second, we utilize the log‐distance pathloss model with RSSs to derive the constraint about distances from the location label to the access points. We construct the optimization problem subject to those constraints to estimate the unwarping transformation. To evaluate the performance of Heimdall, we performed several experiments. Heimdall yields a mean location labeling error of 0.12 m and a standard deviation of location labeling error of 0.06 m for the best case in the experiment. These results demonstrate that a fingerprint‐based indoor positioning system can use the location labels estimated by Heimdall without performance degradation.

Методика формирования цифровых моделей высот геоида для референцных систем координат

A technique was developed for generating digital models of geoid heights for reference coordinate systems using the Trimble Bussines Center software (TBC software). The source models can be the publicly available high-precision global geoid elevation models EGM2008, EIGEN6C4 and other models whose files with the *.ggf extension are compatible with the TBC software. The technique is based on the possibility of exporting orthometric heights from the TBC software in ASCII format to a user-specified area on the surface of the global WGS-84 ellipsoid for a uniform grid of nodes in lati-tude and longitude. The resulting data table is the starting point for calculating geoid heights for reference coordinate systems, which are determined by the parameters of the corresponding reference ellipsoids and Helmert parameters - spatial position, orientation and linear scale relative to the global coordinate system, according to Standard. The initial data are sufficient to calculate the digital model of the geoid heights for an arbitrary reference coordinate system. The expediency of the developed technique lies in the possibility of its use for a much more accurate calculation of the Helmert parameters for regional and other territories compared to the accepted standard values and, accordingly, a more accurate transformation of the three-dimensional positions of points between the general earth and reference coordinate systems (these issues will be discussed in a subsequent publication by the author). Approbation of the technique was carried out for several regional territories: Novosibirsk and Moscow regions, the territory of Western Siberia, the Republic of Belarus in the SK42 and SK95 coordinate systems, and the results are presented graphically in the form of geoid height maps.

Framework to Georeference Point Cloud in GPS Shadowed Urban Environment by Integrating Computer Vision Algorithms and Photogrammetric Techniques

The integration of computer vision algorithms and photogrammetric techniques has become an alternative to the high-cost Mobile Mapping Systems (MMS) and point cloud generation through Structure from Motion (SfM) algorithm is the best example of it. The point cloud generated using SfM is an arbitrary coordinate system and for its georeferencing known global coordinates of the camera exposure stations, rotational and translational parameters are required. The global coordinates of exposure stations are obtained through GNSS (Global Navigation Satellite System). GPS (Global Positioning System) is widely used for getting the positional information of a point. The problem in georeferencing the point cloud arises if the coordinates of a few camera exposure stations are unknown due to GPS shadowing or poor GDOP (Geometric Dilution of Precision). This issue is common in MMS that use laser scanners, GNSS and IMU (inertial measurement unit). In this paper, efforts are made to develop a methodology for handling GPS shadowing or poor accuracy for the georeferencing of arbitrary point clouds generated through SfM. The adopted method uses a blend of photogrammetric techniques of space resection and space intersection to determine the unknown camera exposure stations' coordinates. Bundle adjustment is applied to improve the accuracy of the results obtained. The developed methodology is well analyzed in different cases, and the results show good accuracy in respective cases.

General Relativistic Implicit Monte Carlo Radiation-hydrodynamics

We report on a new capability added to our general relativistic radiation-magnetohydrodynamics code, Cosmos++: an implicit Monte Carlo (IMC) treatment for radiation transport. The method is based on a Fleck-type implicit discretization of the radiation-hydrodynamics equations, but generalized for both Newtonian and relativistic regimes. A multiple reference frame approach is used to geodesically transport photon packets (and solve the hydrodynamics equations) in the coordinate frame, while radiation–matter interactions are handled either in the fluid or electron frames then communicated via Lorentz boosts and orthonormal tetrad bases attached to the fluid. We describe a method for constructing estimators of radiation moments using path-weighting that generalizes to arbitrary coordinate systems in flat or curved spacetime. Absorption, emission, scattering, and relativistic Comptonization are among the matter interactions considered in this report. We discuss our formulations and numerical methods, and validate our models against a suite of radiation and coupled radiation-hydrodynamics test problems in both flat and curved spacetimes.

A conservative implicit-PIC scheme for the hybrid kinetic-ion fluid-electron plasma model on curvilinear meshes

The hybrid kinetic-ion fluid-electron plasma model is widely used to study challenging multi-scale problems in space and laboratory plasma physics. Here, a novel conservative scheme for this model employing implicit particle-in-cell techniques [1] is extended to arbitrary coordinate systems via curvilinear maps from logical to physical space. The scheme features a fully non-linear electromagnetic formulation with a multi-rate time advance – including sub-cycling and orbit-averaging for the kinetic ions. By careful choice of compatible particle-based kinetic-ion and mesh-based fluid-electron discretizations in curvilinear coordinates, as well as particle-mesh interpolations and implicit midpoint time advance, the scheme is proven to conserve total energy for arbitrary curvilinear meshes. In the electrostatic limit, the method is also proven to conserve total momentum for arbitrary curvilinear meshes. Although momentum is not conserved for arbitrary curvilinear meshes in the electromagnetic case, it is for an important subset of Cartesian tensor-packed meshes. The scheme and its novel conservation properties are demonstrated for several challenging numerical problems using different curvilinear meshes, including a merging flux-rope simulation for a space weather application, and a helical m=1 mode simulation for magnetic fusion energy application.

Calculus: High-Dimensional Numerical and Symbolic Calculus in R

The R package calculus implements C++ optimized functions for numerical and symbolic calculus, such as the Einstein summing convention, fast computation of the Levi-Civita symbol and generalized Kronecker delta, Taylor series expansion, multivariate Hermite polynomials, high-order derivatives, ordinary differential equations, differential operators and numerical integration in arbitrary orthogonal coordinate systems. The library applies numerical methods when working with R functions or symbolic programming when working with characters or expressions. The package handles multivariate numerical calculus in arbitrary dimensions and coordinates and implements the symbolic counterpart of the numerical methods whenever possible, without depending on external computer algebra systems. Except for Rcpp, the package has no strict dependencies in order to provide a stable self-contained toolbox that invites re-use.

Efficient Modeling of Hydraulic Fractures with Diamond Boundary in Shale Gas Reservoirs

AbstractThe effects of hydraulic fractures with complex boundaries on the well performance of a shale gas well, considering a more realistic corner point geological model, have rarely been studied previously. In this study, the nonintrusive embedded discrete fracture model (EDFM) method was employed to investigate the effects of different boundary shapes of hydraulic fracture, coupling a network of sophisticated natural fractures discrete fracture network (DFN) on well performance. First, by implementing the powerful EDFM technology, concepts of two categories (rectangle and diamond) of hydraulic fracture with different boundaries were designed. Next, the geometric equations defining vertices of multiple rectangular- or diamond-shaped hydraulic fractures in arbitrary coordinate systems were derived. Subsequently, the horizontal well with multistaged hydraulic fractures and sophistically oriented 3D natural fractures was inputted into the reservoir model to perform history matching. After history matching, the results were further analyzed to compare the production forecast from the two categories. The results show that 20-year cumulative gas productions for rectangle- and diamond-shaped fractures are approximately 1.237×108 m3 and 1.486×108 m3, respectively. In other words, the diamond category can produce 20.1% more gas than the rectangle category. For cumulative water production, the diamond category produces 3.8×104 m3, as against the 3.0×104 m3 produced by the rectangle category (or 26.7% more). This implies that the diamond-shaped fractures have the potential to reach the far field region of the reservoir away from the wellbore. This means that more intersections with natural fractures DFN can be achieved, and more drainage area is unlocked. The visualization of pressure distributions and drainage volume was easily shown, and these results further confirm that the extent of fluid drainage by the diamond fracture is larger compared to that by the rectangular fracture given the same total surface area.

Vibration and damping characteristics of CNTR viscoelastic skewed shell structures under the influence of hygrothermal conditions

The vibration and damping characteristics of carbon nanotubes reinforced (CNTR) skewed shell structure under a hygrothermal environment have been investigated using the finite element method. CNT as reinforcing phase and polymer as matrix phase are considered for the nanocomposites (NCs) based viscoelastic skewed shell structure. Dynamic mechanical analysis is used to conduct the creep test for NCs samples which were fabricated, as per ASTM-D4065 standard, and obtained the viscoelastic properties in the frequency domain under different hygrothermal conditions. The shell geometry is defined by considering an arbitrary coordinate system for the skewed shell structure. Finite element modelling has been done with Serendipity element with five degrees of freedom in all eight nodes. The present formulation is based on Koiter’s shell theory and first-order shear deformation theory is considered to incorporate the transverse shear effect based on Mindlin’s hypothesis. The frequency dependant viscoelastic properties are directly used to obtain the frequency responses of the skewed shell panel using fast Fourier transform (FFT) whereas the transient responses are determined using inverse fast Fourier transform (IFFT). An in-house MATLAB code is developed for the numerical simulation and the accuracy of the proposed formulation is validated with available results in literatures and using ANSYS software. A parametric study has been carried out for the skewing angle and CNT volume fraction on the vibration behaviour of different thin and thick NC skewed shell structures under various hygrothermal conditions.

Analysis of wave scattering from 2D curved metasurfaces using Floquet and Fourier series expansions

An efficient technique for calculating the scattering from curved metasurfaces using the extinction theorem in conjunction with the Floquet and Fourier series expansions is presented. Here, we treat the two-dimensional metasurfaces that have transversal polarizabilities with no variation along the y-axis. The boundary conditions at the metasurface are given by the generalized sheet transition conditions (GSTCs) whose susceptibilities are given in an arbitrary local coordinate system. First, we use the extinction theorem to provide integral equations of the scattering problem. The integral equations involve the Green's functions, tangential electric and magnetic fields and their normal derivatives in regions above and below the metasurface. Then, we employ the Floquet theorem that gives us the analytical periodic Green's functions of each region. Next, we employ the Fourier theorem to expand the tangential fields in terms of unknown Fourier coefficients. The GSTCs and the integral equations provide equations to be solved for the unknowns. The method can calculate scattering from both periodic and non-periodic metasurfaces. The technique is used to analyse different applied problems such as carpet cloaking, illusion, and radar echo width reduction. The method is fast and accurate and can efficiently treat metasurfaces with electrically large curved geometries with dimensions as large as 120 times the wavelength.

Numerical solution of compressible Euler and Magnetohydrodynamic flow past an infinite cone

A numerical scheme is developed for systems of conservation laws on manifolds which arise in high speed aerodynamics and magneto-aerodynamics. The systems are presented in an arbitrary coordinate system on the manifold and involve source terms which account for the curvature of the domain. In order for a numerical method to accurately capture the behavior of the system it is solving, the equations must be discretized in a way that is not only consistent in value, but also models the appropriate character of the system. Such a discretization is presented in this work which preserves the tensorial transformation relationships involved in formulating equations in a curved space. A numerical method is then developed and applied to the conical Euler and Ideal Magnetohydrodynamic equations. To the authors’ knowledge, this is the first demonstration of a numerical solver for the conical Ideal MHD equations. Along with the curvature of the domain, is the added challenge that both systems of equations are of mixed type. Both systems change between elliptic and hyperbolic type throughout the domain, which had to be accommodated by the numerical method.

Stress–temperature equations of motion of Ignaczak and Beltrami–Michell types in arbitrary curve coordinate system

يبيًّن البحث بالنموذج الرياضي لجسم مرن متماثل المناجي Isotropic)) ومتجانس Homogeneous))، ومهمل البنية الجزيئية ويعاني من انفعالات مرنة لامتناهية في الصغر، والذي يُرمز له اختصاراً بـ (H)، وسيتم مناقشة الشكل التنسوري الناطق في نظام إحداثي منحني كيفي، لكلٍ من النموذج الرياضي التقليدي ونموذج لامي الرياضي لجسم هوك المرن (H)، المًعتبر، في حالته التحريكية، بوجود حرارة. سيتم مناقشة الشكل التنسوري الناطق في نظام إحداثي منحني كيفي على المتنوعة التقليدية ثلاثية الأبعاد، لكل من معادلات بيلترامي- ميشيل المعممة ومعادلات إغناتشاك بالإجهادات والحرارة، ومسألة الاتمام المتعلقة بها، وذلك لأجل الجسم الثرموديناميكي المرن (H) الخاضع للحرارة.

CNNs on surfaces using rotation-equivariant features

This paper is concerned with a fundamental problem in geometric deep learning that arises in the construction of convolutional neural networks on surfaces. Due to curvature, the transport of filter kernels on surfaces results in a rotational ambiguity, which prevents a uniform alignment of these kernels on the surface. We propose a network architecture for surfaces that consists of vector-valued, rotation-equivariant features. The equivariance property makes it possible to locally align features, which were computed in arbitrary coordinate systems, when aggregating features in a convolution layer. The resulting network is agnostic to the choices of coordinate systems for the tangent spaces on the surface. We implement our approach for triangle meshes. Based on circular harmonic functions, we introduce convolution filters for meshes that are rotation-equivariant at the discrete level. We evaluate the resulting networks on shape correspondence and shape classifications tasks and compare their performance to other approaches.