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8 results for “geodesic”
Geodesic-BP Dataset
<p>This dataset contains the results of our method <i>Geodesic-BP</i>, presented in <a href="https://arxiv.org/abs/2308.08410">https://arxiv.org/abs/2308.08410</a>. <br>The original rabbit torso model can be found in <a href="https://zenodo.org/record/6340066">https://zenodo.org/record/6340066</a>, on which we based the setup.</p><p>The dataset consists of the final results, optimization results over the 400 iterations and a pseudo-bidomain simulation from the final result. All files are provided in variants of the VTK file format (<a href="https://vtk.org/">https://vtk.org/</a>) The setup/result files are organized as follows:</p><ul><li>result_mesh_init_final.vtu - The biventricular mesh containing both initial and final solution φk</li><li>result_x0_init.vtp - The initial conditions (xi , ti) used in the first optimization iteration</li><li>result_x0_final.vtp - The initial conditions (xi, ti) computed using our optimization algorithm</li><li>ecgs.npz - Numpy-readable (np.load) arrays of ECGs (ecg_init, ecg_final, ecg_target)</li><li>ecgs.vtp - Target and optimized ECGs converted to a Paraview-readable format</li><li>ecg_history.npz - Numpy-readable array of the ECGs over the iterations</li></ul><p>The animation files present allows you to preview the solution in each iteration </p><ul><li>phi_history.xdmf - The solution φk in each iterations (surface only)</li><li>x0_history.xdmf - The initial conditions in each iteration</li><li>ecg_anim.xdmf - The computed ECGs in each iteration</li></ul><p>The files can be easily viewed in VTK-compatible viewers, such as Paraview (<a href="https://www.paraview.org/">https://www.paraview.org/</a>). We additionally provide a Paraview state file (preview.pvsm), which when opened in Paraview automatically creates several views that visualize the data in different views. Simply open Paraview, select File -> Load State, locate the preview.pvsm. In the next prompt (Load State Options) select "Search files under specified directory" and locate the folder with the files, then press OK.</p>
Test set for geodesics
<p>This is a set of 500000 geodesics for the WGS84 ellipsoid; this is an ellipsoid of revolution with equatorial radius <em>a</em> = 6378137 m and flattening <em>f</em> = 1/298.257223563.</p> <p>Each line of the test set gives 10 space delimited numbers</p> <ul> <li>latitude at point 1, φ<sub>1</sub> (degrees, exact)</li> <li>longitude at point 1, λ<sub>1</sub> (degrees, always 0)</li> <li>azimuth at point 1, α<sub>1</sub> (clockwise from north in degrees, exact)</li> <li>latitude at point 2, φ<sub>2</sub> (degrees, accurate to 10<sup>−18</sup> deg)</li> <li>longitude at point 2, λ<sub>2</sub> (degrees, accurate to 10<sup>−18</sup> deg)</li> <li>azimuth at point 2, α<sub>2</sub> (degrees, accurate to 10<sup>−18</sup> deg)</li> <li>geodesic distance from point 1 to point 2, <em>s</em><sub>12</sub> (meters, exact)</li> <li>arc distance on the auxiliary sphere, σ<sub>12</sub> (degrees, accurate to 10<sup>−18</sup> deg)</li> <li>reduced length of the geodesic, <em>m</em><sub>12</sub> (meters, accurate to 0.1 pm)</li> <li>the area between the geodesic and the equator, <em>S</em><sub>12</sub> (m<sup>2</sup>, accurate to 1 mm<sup>2</sup>)</li> </ul> <p>These are computed using high-precision direct geodesic calculations with the given φ<sub>1</sub>, λ<sub>1</sub>, α<sub>1</sub>, and <em>s</em><sub>12</sub>. The distance <em>s</em><sub>12</sub> always corresponds to an arc length σ<sub>12</sub> ≤ 180°, so the given geodesics give the shortest paths from point 1 to point 2. For simplicity and without loss of generality, φ<sub>1</sub> is chosen in [0°, 90°], λ<sub>1</sub> is taken to be zero, α<sub>1</sub> is chosen in [0°, 180°]. Furthermore, φ<sub>1</sub> and α<sub>1</sub> are taken to be multiples of 10<sup>−12</sup> deg and <em>s</em><sub>12</sub> is a multiple of 0.1 μm in [0 m, 20003931.4586254 m]. This results in λ<sub>2</sub> in [0°, 180°] and α<sub>2</sub> in [0°, 180°].</p> <p>The contents of the file are as follows:</p> <ul> <li>100000 entries randomly distributed</li> <li>50000 entries which are nearly antipodal</li> <li>50000 entries with short distances</li> <li>50000 entries with one end near a pole</li> <li>50000 entries with both ends near opposite poles</li> <li>50000 entries which are nearly meridional</li> <li>50000 entries which are nearly equatorial</li> <li>50000 entries running between vertices (α<sub>1</sub> = α<sub>2</sub> = 90°)</li> <li>50000 entries ending close to vertices</li> </ul> <p>The values for <em>s</em><sub>12</sub> for the geodesics running between vertices are truncated to a multiple of 0.1 pm and this is used to determine point 2.</p>
Test set of geodesics on a triaxial ellipsoid
<p>This is a set of 500000 shortest geodesics on a triaxial ellipsoid. The ellipsoid is defined by</p> <p>$$\frac{X^2}{a^2} + \frac{Y^2}{b^2} + \frac{Z^2}{c^2} - 1 = 0,$$</p> <p>with \(a = \sqrt2\), \(b = 1\), \(c = 1/\sqrt2\) (measured in arbitrary units). (This ellipsoid was studied by A. Cayley, <em>On the geodesic lines on an ellipsoid</em>, Mem. Roy. Astron. Soc. <strong>39</strong>, 31-53, 1872.) Each line of the test set consists of 10 space-delimited numbers</p> <ul> <li>the latitude at point 1, \(\beta_1\) (\(^\circ\), exact)</li> <li>the longitude at point 1, \(\omega_1\) (\(^\circ\), exact)</li> <li>the azimuth at point 1, \(\alpha_1\) (\(^\circ\), accurate to \(10^{-18}{}^\circ\))</li> <li>the latitude at point 2, \(\beta_2\) (\(^\circ\), exact)</li> <li>the longitude at point 2, \(\omega_2\) (\(^\circ\), exact)</li> <li>the azimuth at point 2, \(\alpha_2\) (\(^\circ\), accurate to \(10^{-18}{}^\circ\))</li> <li>the geodesic distance from 1 to 2, \(s_{12}\) (units, accurate to \(10^{-20}\))</li> <li>the reduced length of the geodesic, \(m_{12}\) (units, accurate to \(10^{-20}\))</li> <li>the geodesic scale, \(M_{12}\) (accurate to \(10^{-20}\))</li> <li>the geodesic scale, \(M_{21}\) (accurate to \(10^{-20}\))</li> </ul> <p>Here \(\beta\), \(\omega\), and \(\alpha\), are the <em>ellipsoidal</em> latitude, longitude, and azimuth. For a given \((\beta, \omega)\), the Cartesian coordinates of a point are</p> <p>$$\begin{align}<br> X &= a \cos\omega<br> \frac{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}}<br> {\sqrt{a^2 - c^2}}, \\<br> Y &= b \cos\beta \sin\omega, \\<br> Z &= c \sin\beta<br> \frac{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}}<br> {\sqrt{a^2 - c^2}}.<br>\end{align}$$</p> <p>Lines of constant \(\beta\) and \(\omega\) are orthogonal. The azimuth \(\alpha\) of a geodesic is the direction measured clockwise from North (defined as \(\beta\) increasing at constant \(\omega\)). The coordinates are singular at the four <em>umbilical</em> points \(\cos\beta = \sin\omega = 0\). The azimuth of a geodesic jumps by \(\pm\frac12\pi\) on passage through such points and the value for such points is the azimuth on <em>leaving</em> the umbilical point.</p> <p>The geodesics are computed using high-precision inverse calculations with the exact <em>integer</em> values for \((\beta_1, \omega_1)\) and \((\beta_2, \omega_1)\). Any of the other entries reported as an integer is also exact.</p> <p>For most pairs of points, there is a unique shortest geodesic. However</p> <ul> <li>for opposite umbilical points, \(\alpha_1\) and \(\alpha_2\) can take on arbitrary values provided that the ratio \(\tan\alpha_1/\tan\alpha_2\) is maintained;</li> <li>if \(\beta_1 + \beta_2 = 0\) and if \(\cos\alpha_1\) and \(\cos\alpha_2\) have opposite signs, then there is another shortest geodesic with azimuths \(\pi - \alpha_1\) and \(\pi - \alpha_2\).</li> </ul> <p>For a particular \((\beta_1, \omega_1)\) and \((\beta_2, \omega_2)\), additional geodesics of the same length can be trivially generated by swapping the points or by reflecting them in any of the coordinate planes. A non-trivial symmetry is given by swapping <em>just</em> the longitude coordinates; this also results in a geodesic of the same length. The data set has had any such redundant geodesics removed.</p> <p>The data set is sorted according to whether either point</p> <ul> <li>is an umbilical point</li> <li>lies on the middle principal ellipse, with \(\sin\omega = 0\)</li> <li>lies on the middle principal ellipse, with \(\cos\beta = 0\)</li> <li>lies on the major principal ellipse, \(Z = 0\)</li> <li>lies on the minor principal ellipse, \(X = 0\)</li> <li>is near an umbilical point</li> <li>is general (all other points)</li> </ul> <p>Approximately 85% of the entries are with two general points. If only a small set of random test cases is needed, select a random subset with, e.g.,</p> <p> <code>shuf Geod3Test.txt | head -1000 > Geod3Test-samp.txt</code></p> <p> </p>
Accompanying dataset for the paper "Mesh Density and Geodesic Tortuosity in Planar Triangular Tesselations Devoted to Fracture Mechanics"
<h2>Contributions</h2> <ul> <li>Author #1 is the major contributor to the paper writing. Author #1 has created most of the Figures in the paper and has provided all the results concerning the geodesic tortuosity of planar triangular meshes.</li> <li>Author #2 and Author #3 have provided the results concerning the density of planar triangular meshes.</li> </ul> <h2>Funding sources</h2> <ul> <li>This work was funded by the French Institute for Radiation Protection and Nuclear Safety (IRSN) and the University of Montpellier, France.</li> </ul> <h2>Data structure and information</h2> <p>All the results and figures presented in the paper have been obtained with a python script available in the <code>workflows</code> folder. </p> <p>Data files processed by that script are provided in the <code>data</code> folder. The pickle files <code>t_{nb}.pkl</code> inside the <code>data/pkl</code> folder contain lists of tortuosity values computed on a <code>gmsh</code> mesh for several paths containing various number of edges (<code>nb</code>).</p> <p><code>pkl</code>files are produced via a function call to <code>main.py:comp_real_tortuosities(nb)</code></p> <h2>Data structure and information</h2> <ul> <li><code>workflows/</code> - folder containing plotting scripts<ul> <li><code>reproduce.sh</code> - bash script launching python and compressing figures</li> <li><code>main.py</code> - python script for figure creation.</li> <li><code>/utils</code> - folder containing additional python scripts for figures creation</li> <li><code>/figures</code> - folder containing the figures produced by the python script</li> </ul> </li> <li><code>data/</code> - data folder<ul> <li><code>gmsh</code> - folder containing mesh in gmsh information</li> <li><code>pkl</code> - folder containing the pickled vectors "t_{nb}.pkl".</li> </ul> </li> </ul> <h2>Paper Description</h2> <p>In fracture mechanics, the mesh sensitivity is a key issue. It is particularly true concerning cohesive volumetric finite element methods in which the crack path and the overall behavior are respectively influenced by the mesh topology and the mesh density. Poisson-Delaunay tessellations parameters, including the edge length distributions, were widely studied in the literature but very few works concern the mesh density and topology in Delaunay type meshes suitable for finite element simulations, which is of crucial interest for practical use.Starting from previous results concerning Poisson-Delaunay tessellations and studying in detail the Lloyd relaxation algorithm, we propose estimates for the probability density functions of the edge length and triangle top angles sets. These estimates depend both on the intensity of the underlying point process and on an efficiency index associated to the global quality of the mesh. The global and local accuracies of these estimates are checked for various standard mesh generators. Finally the mesh density and geodesic tortuosity are estimated for standard random or structured triangular meshes typically used in finite element simulations.These results provide practical formulas to estimate bias introduced by the mesh density and topology onthe results of cohesive-volumetric finite element simulations.</p>
Improving Neutral Density Predictions Using Exospheric Temperatures Calculated on a Geodesic, Polyhedral Grid
<p>These files provide supplemental data, additional graphs of neutral density predictions, and basic program code and required files, to accompany the paper "Improving Neutral Density Predictions Using Exospheric Temperatures Calculated on a Geodesic, Polyhedral Grid,” published in the journal <em>Space Weather, </em>doi:10.1029/2019SW002355. For more information, refer to the READ_ME.pdf documentation and the journal paper.</p>
Precomputed results for Geodesic Slice Sampling on the Sphere (GeoSSS) paper
<p>Precomputed results used in the publication titled "<a href="https://arxiv.org/abs/2301.08056">Geodesic Slice Sampling on the Sphere</a>". This can be used with our package on Github: <a href="https://github.com/microscopic-image-analysis/geosss">https://github.com/microscopic-image-analysis/geosss</a></p> <p>Follow the below steps:</p> <p>1. Download and unzip all the files. </p> <p>2. Clone the github repository and navigate to the root directory.</p> <pre><code>git clone https://github.com/microscopic-image-analysis/geosss.git cd geosss</code></pre> <p>3. Move the data to a new <em>"results"</em> directory inside the repository. After unzipping folders from "<em>mixture_vMF_d10_K5_kappa50" </em>to <em>"mixture_vMF_d10_K5_kappa500" </em>(10 folders), please move them to a new <em>"results/mix_vMF_d10_K5"</em> directory.</p> <p>4. You can use the "geosss/scripts" directory to use these precomputed results and plot them, see the info on the <a href="https://github.com/microscopic-image-analysis/geosss/tree/main">GitHub repository</a> for installing the package. </p>
Supplementary data for paper "Extreme mass-ratio inspiral and waveforms for a spinning body into a Kerr black hole via osculating geodesics and near-identity transformations"
<p>Radiation-reaction fluxes and NIT interpolant data for paper "Extreme mass-ratio inspiral and waveforms for a spinning body into a Kerr black hole via osculating geodesics and near-identity transformations"</p>
Geodesic
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