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Journal articles on the topic 'Geodesic distances'

Author: Grafiati

Published: 14 December 2024

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1

Li, Yue, Logan Numerow, Bernhard Thomaszewski, and Stelian Coros. "Differentiable Geodesic Distance for Intrinsic Minimization on Triangle Meshes." ACM Transactions on Graphics 43, no. 4 (July 19, 2024): 1–14. http://dx.doi.org/10.1145/3658122.

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Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

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2

Guzmán Naranjo, Matías, and Gerhard Jäger. "Euclide, the crow, the wolf and the pedestrian: distance metrics for linguistic typology." Open Research Europe 3 (June 21, 2023): 104. http://dx.doi.org/10.12688/openreseurope.16141.1.

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It is common for people working on linguistic geography, language contact and typology to make use of some type of distance metric between lects. However, most work so far has either used Euclidean distances, or geodesic distance, both of which do not represent the real separation between communities very accurately. This paper presents two datasets: one on walking distances and one on topographic distances between over 8700 lects across all macro-areas. We calculated walking distances using Open Street Maps data, and topographic distances using digital elevation data. We evaluate these distances. We evaluate these distance metrics on three case studies and show that topographic distance tends to outperform the other distance metrics, but geodesic distances can be used as an adequate approximation in some cases.

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3

Mejia-Parra, Daniel, Jairo R. Sánchez, Jorge Posada, Oscar Ruiz-Salguero, and Carlos Cadavid. "Quasi-Isometric Mesh Parameterization Using Heat-Based Geodesics and Poisson Surface Fills." Mathematics 7, no. 8 (August 17, 2019): 753. http://dx.doi.org/10.3390/math7080753.

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In the context of CAD, CAM, CAE, and reverse engineering, the problem of mesh parameterization is a central process. Mesh parameterization implies the computation of a bijective map ϕ from the original mesh M ∈ R 3 to the planar domain ϕ ( M ) ∈ R 2 . The mapping may preserve angles, areas, or distances. Distance-preserving parameterizations (i.e., isometries) are obviously attractive. However, geodesic-based isometries present limitations when the mesh has concave or disconnected boundary (i.e., holes). Recent advances in computing geodesic maps using the heat equation in 2-manifolds motivate us to revisit mesh parameterization with geodesic maps. We devise a Poisson surface underlying, extending, and filling the holes of the mesh M. We compute a near-isometric mapping for quasi-developable meshes by using geodesic maps based on heat propagation. Our method: (1) Precomputes a set of temperature maps (heat kernels) on the mesh; (2) estimates the geodesic distances along the piecewise linear surface by using the temperature maps; and (3) uses multidimensional scaling (MDS) to acquire the 2D coordinates that minimize the difference between geodesic distances on M and Euclidean distances on R 2 . This novel heat-geodesic parameterization is successfully tested with several concave and/or punctured surfaces, obtaining bijective low-distortion parameterizations. Failures are registered in nonsegmented, highly nondevelopable meshes (such as seam meshes). These cases are the goal of future endeavors.

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4

WANG, SONGJING, ZHOUYU YU, and LIFENG XI. "AVERAGE GEODESIC DISTANCE OF SIERPINSKI GASKET AND SIERPINSKI NETWORKS." Fractals 25, no. 05 (September 4, 2017): 1750044. http://dx.doi.org/10.1142/s0218348x1750044x.

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The average geodesic distance is concerned with complex networks. To obtain the limit of average geodesic distances on growing Sierpinski networks, we obtain the accurate value of integral in terms of average geodesic distance and self-similar measure on the Sierpinski gasket. To provide the value of integral, we find the phenomenon of finite pattern on integral inspired by the concept of finite type on self-similar sets with overlaps.

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5

Jenčová, Anna. "Geodesic distances on density matrices." Journal of Mathematical Physics 45, no. 5 (May 2004): 1787–94. http://dx.doi.org/10.1063/1.1689000.

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6

Guzmán Naranjo, Matías, and Gerhard Jäger. "Euclide, the crow, the wolf and the pedestrian: distance metrics for linguistic typology." Open Research Europe 3 (July 2, 2024): 104. http://dx.doi.org/10.12688/openreseurope.16141.2.

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It is common for people working on linguistic geography, language contact and typology to make use of some type of distance metric between lects. However, most work so far has either used Euclidean distances, or geodesic distance, both of which do not represent the real separation between communities very accurately. This paper presents two datasets: one on walking distances and one on topographic distances between over 8700 lects across all macro-areas. We calculated walking distances using Open Street Maps data, and topographic distances using digital elevation data. We evaluate these distance metrics on three case studies and show that from the four distances, the topographic and geodesic distances showed the most consistent performance across datasets, and would be likely to be reasonable first choices. At the same time, in most cases, the Euclidean distances were not much worse than the other distances, and might be a good enough approximation in cases for which performance is critical, or the dataset cover very large areas, and the point-location information is not very precise.

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7

BORGELT, MAGDALENE G., MARC VAN KREVELD, and JUN LUO. "GEODESIC DISKS AND CLUSTERING IN A SIMPLE POLYGON." International Journal of Computational Geometry & Applications 21, no. 06 (December 2011): 595–608. http://dx.doi.org/10.1142/s0218195911003822.

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Let P be a simple polygon of n vertices and let S be a set of N points lying in the interior of P. A geodesic diskGD(p,r) with center p and radius r is the set of points in P that have a geodesic distance ≤ r from p (where the geodesic distance is the length of the shortest polygonal path connection that lies in P). In this paper we present an output sensitive algorithm for finding all N geodesic disks centered at the points of S, for a given value of r. Our algorithm runs in [Formula: see text] time, for some constant c and output size k. It is the basis of a cluster reporting algorithm where geodesic distances are used.

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8

Hino, Masanori. "Geodesic Distances and Intrinsic Distances on Some Fractal Sets." Publications of the Research Institute for Mathematical Sciences 50, no. 2 (2014): 181–205. http://dx.doi.org/10.4171/prims/129.

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9

Mahdi, Hussein Alwan. "A MODIFIED METHOD FOR DETERMINATION OF SCALE FACTOR OF THE PROJECTED GEODESIC." Journal of Engineering 12, no. 03 (September 1, 2006): 882–95. http://dx.doi.org/10.31026/j.eng.2006.03.31.

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Conformal projection is one of the most important aspects that geodesy dealing with. Thedetermination of the scale factors in the meridian, the parallel and projected geodesic directions are thefinal result of the conformal projection. Methods for determining the scale factors in the meridian andthe parallel directions have a quite sufficient accuracy. While methods for determining the projectedgeodesic have different accuracy and computation complicity.This research adopts a modified method for computing the exact value of scale factor ingeodesic direction. In this method the scale factor is obtained by determining the true and projecteddistances of the geodetic line. In the traditional methods for determining the projected distance it isusual to use the 1/3 Simpson's rule in the computations while the modified method the 3/8 Simpson'srule is used.Computations and mathematical tests were carried out to obtain the scale factors using thetraditional methods and comparison was made with modified method.By applying the developed method and the traditional methods to calculate the scale factor, it wasfound that the modified method is more accurate and the projected distances can be obtained exactly.

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10

Noyel, Guillaume, Jesús Angulo, and Dominique Jeulin. "FAST COMPUTATION OF ALL PAIRS OF GEODESIC DISTANCES." Image Analysis & Stereology 30, no. 2 (June 30, 2011): 101. http://dx.doi.org/10.5566/ias.v30.p101-109.

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Computing an array of all pairs of geodesic distances between the pixels of an image is time consuming. In the sequel, we introduce new methods exploiting the redundancy of geodesic propagations and compare them to an existing one. We show that our method in which the source point of geodesic propagations is chosen according to its minimum number of distances to the other points, improves the previous method up to 32 % and the naive method up to 50 % in terms of reduction of the number of operations.

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11

Owen, Megan. "Computing Geodesic Distances in Tree Space." SIAM Journal on Discrete Mathematics 25, no. 4 (January 2011): 1506–29. http://dx.doi.org/10.1137/090751396.

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12

Ambjørn, J., and T. G. Budd. "Geodesic distances in Liouville quantum gravity." Nuclear Physics B 889 (December 2014): 676–91. http://dx.doi.org/10.1016/j.nuclphysb.2014.10.029.

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13

Menéndez, M. L., D. Morales, L. Pardo, and M. Salicrú. "Statistical tests based on geodesic distances." Applied Mathematics Letters 8, no. 1 (January 1995): 65–69. http://dx.doi.org/10.1016/0893-9659(94)00112-p.

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14

Du, Mingjing, Shifei Ding, Xiao Xu, and Yu Xue. "Density peaks clustering using geodesic distances." International Journal of Machine Learning and Cybernetics 9, no. 8 (March 2, 2017): 1335–49. http://dx.doi.org/10.1007/s13042-017-0648-x.

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15

Wang, Xiaoning, Zheng Fang, Jiajun Wu, Shi-Qing Xin, and Ying He. "Discrete geodesic graph (DGG) for computing geodesic distances on polyhedral surfaces." Computer Aided Geometric Design 52-53 (March 2017): 262–84. http://dx.doi.org/10.1016/j.cagd.2017.03.010.

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16

DENG, JUAN, and QIN WANG. "ASYMPTOTIC FORMULA OF AVERAGE DISTANCES ON FRACTAL NETWORKS MODELED BY SIERPINSKI TETRAHEDRON." Fractals 27, no. 07 (November 2019): 1950120. http://dx.doi.org/10.1142/s0218348x19501202.

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This paper concerns the average distances of evolving networks modeled by Sierpinski tetrahedron. We express the limit of average distances on reorganized networks as an integral of geodesic distance on Sierpinski tetrahedron. Based on the self-similarity and renewal theorem, we obtain the asymptotic formula on the average distance of our evolving networks.

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17

Luo, Songting, Shingyu Leung, and Jianliang Qian. "An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria." Communications in Computational Physics 10, no. 5 (November 2011): 1113–31. http://dx.doi.org/10.4208/cicp.020210.311210a.

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AbstractThe equilibrium metric for minimizing a continuous congested traffic model is the solution of a variational problem involving geodesic distances. The continuous equilibrium metric and its associated variational problem are closely related to the classical discrete Wardrop’s equilibrium. We propose an adjoint state method to numerically approximate continuous traffic congestion equilibria through the continuous formulation. The method formally derives an adjoint state equation to compute the gradient descent direction so as to minimize a nonlinear functional involving the equilibrium metric and the resulting geodesic distances. The geodesic distance needed for the state equation is computed by solving a factored eikonal equation, and the adjoint state equation is solved by a fast sweeping method. Numerical examples demonstrate that the proposed adjoint state method produces desired equilibrium metrics and outperforms the subgradient marching method for computing such equilibrium metrics.

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18

Gienko, Elena G., Alexander V. Elagin, and Konstantin Yu Reznichenko. "RESULTS OF BUILDING A LOCAL QUASIGEOID MODEL ON THE TERRITORY OF THE GEODETIC TRAINING GROUND OF SSUGT." Interexpo GEO-Siberia 1 (May 21, 2021): 252–60. http://dx.doi.org/10.33764/2618-981x-2021-1-252-260.

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The results of building a local quasigeoid model by various methods on the territory of the geodesic training ground of SSUGT, based on the data of geometric leveling, GNSS measurements, gravimetry and astronomical measurements, are presented. The advantages of using a two-dimensional model of a quasigeoid in ellipsoidal coordinates over the "flat model" of height calibration widely used in GNSS technologies are shown. The criteria for choosing a method for building a quasigeoid model on a local territory and criteria for evaluating the quality of the results are determined. The results of determining the deviations of the vertical line in a given area, with control according to astronomo-geodesic measurements, are presented. In particular, a method for quick determining the deviations of a vertical line from the differences in astronomical and geodetic zenith distances was tested. A conclusion about the best method for determining the parameters of the local model of the quasigeoid and the deviations of the vertical line for a given territory is made. The results of the research are of practical significance for the training of students and specialists in the field of geodesy.

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19

Chen, Shuangmin, Nailei Hei, Shun Hu, Zijia Yue, and Ying He. "Convex Quadratic Programming for Computing Geodesic Distances on Triangle Meshes." Mathematics 12, no. 7 (March 27, 2024): 993. http://dx.doi.org/10.3390/math12070993.

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Querying the geodesic distance field on a given smooth surface is a fundamental research pursuit in computer graphics. Both accuracy and smoothness serve as common indicators for evaluating geodesic algorithms. In this study, we argue that ensuring that the norm of the triangle-wise estimated gradients is not larger than 1 is preferable compared to the widely used eikonal condition. Inspired by this, we formulate the geodesic distance field problem as a Quadratically Constrained Linear Programming (QCLP) problem. This formulation can be further adapted into a Quadratically Constrained Quadratic Programming (QCQP) problem by incorporating considerations for smoothness requirements. Specifically, when enforcing a Hessian-energy-based smoothing term, our formulation, named QCQP-Hessian, effectively mitigates the cusps in the geodesic isolines within the near-ridge area while maintaining accuracy in the off-ridge area. We conducted extensive experiments to demonstrate the accuracy and smoothness advantages of QCQP-Hessian.

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20

XI, LIFENG, QIANQIAN YE, and JIANGWEN GU. "AVERAGE GEODESIC DISTANCE OF NODE-WEIGHTED SIERPINSKI NETWORKS." Fractals 27, no. 07 (November 2019): 1950110. http://dx.doi.org/10.1142/s0218348x1950110x.

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This paper discusses the asymptotic formula of average distances on node-weighted Sierpinski skeleton networks by using the integral of geodesic distance in terms of self-similar measure on the Sierpinski gasket with respect to the weight vector.

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21

Scheffer, Christian, and Jan Vahrenhold. "Approximating geodesic distances on 2-manifolds inR3." Computational Geometry 47, no. 2 (February 2014): 125–40. http://dx.doi.org/10.1016/j.comgeo.2012.05.001.

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22

De Sanctis, Angela A., Stefano A. Gattone, and Fotios D. Oikonomou. "Alpha geodesic distances for clustering of shapes." Results in Applied Mathematics 18 (May 2023): 100363. http://dx.doi.org/10.1016/j.rinam.2023.100363.

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23

ZHU, JIALI, LI TIAN, and QIN WANG. "AVERAGE GEODESIC DISTANCE ON SIERPINSKI HEXAGON AND SIERPINSKI HEXAGON NETWORKS." Fractals 27, no. 05 (August 2019): 1950077. http://dx.doi.org/10.1142/s0218348x19500774.

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In this paper, we investigate the average geodesic distance on the Sierpinski hexagon in terms of finite patterns on integrals. Applying this result, we also obtain the asymptotic formula for average distances of Sierpinski hexagon networks.

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24

He, Vivian. "Equivalent topologies on the contracting boundary." Glasnik Matematicki 58, no. 1 (June 30, 2023): 75–83. http://dx.doi.org/10.3336/gm.58.1.06.

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The contracting boundary of a proper geodesic metric space generalizes the Gromov boundary of a hyperbolic space. It consists of contracting geodesics up to bounded Hausdorff distances. Another generalization of the Gromov boundary is the \(\kappa\)–Morse boundary with a sublinear function \(\kappa\). The two generalizations model the Gromov boundary based on different characteristics of geodesics in Gromov hyperbolic spaces. It was suspected that the \(\kappa\)–Morse boundary contains the contracting boundary. We will prove this conjecture: when \(\kappa =1\) is the constant function, the 1-Morse boundary and the contracting boundary are equivalent as topological spaces.

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25

Kadaj, Roman. "Empirical methods of reducing the observations in geodetic networks." Geodesy and Cartography 65, no. 1 (June 1, 2016): 13–40. http://dx.doi.org/10.1515/geocart-2016-0001.

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Abstract The paper presents empirical methodology of reducing various kinds of observations in geodetic network. A special case of reducing the observation concerns cartographic mapping. For numerical illustration and comparison of methods an application of the conformal Gauss-Krüger mapping was used. Empirical methods are an alternative to the classic differential and multi-stages methods. Numerical benefits concern in particular very long geodesics, created for example by GNSS vectors. In conventional methods the numerical errors of reduction values are significantly dependent on the length of the geodesic. The proposed empirical methods do not have this unfavorable characteristics. Reduction value is determined as a difference (or especially scaled difference) of the corresponding measures of geometric elements (distances, angles), wherein these measures are approximated independently in two spaces based on the known and corresponding approximate coordinates of the network points. Since in the iterative process of the network adjustment, coordinates of the points are systematically improved, approximated reductions also converge to certain optimal values.

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Yang, Dongsheng, Ting Li, Bo Hu, Jing Gao, and Chunsheng Wang. "Multimode Process Monitoring Based on Geodesic Distance." International Journal of Software Engineering and Knowledge Engineering 28, no. 09 (September 2018): 1225–48. http://dx.doi.org/10.1142/s0218194018400132.

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A novel monitoring strategy is proposed for multimode process in which mode clustering and fault detection based on geodesic distance (GD) are integrated. To start with, the empowered adjacency matrix of normalized training dataset is obtained and improved Dijkstra algorithm (IDA) is utilized to calculate the geodesic distance between each sample data so as to characterize the shortest distance of the nonlinear data within local areas accurately. Next, GD matrix algorithm is presented as an optimal clustering solution for a multimode process dataset. Then, the GDS model is established in each operating mode. Monitoring statistics based on the power of geodesic distance are structured based on square sum of Euclidean distances. Once the test data is detected as fault data, mode location based on deviation coefficient is conducted to narrow the scope of the inspection fault. Finally, the validity and usefulness of the proposed GDMPM monitoring method are demonstrated through the Tennessee Eastman (TE) benchmark process.

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27

FAN, JIAQI, JIANGWEN GU, LIFENG XI, and QIN WANG. "AVERAGE DISTANCES OF A FAMILY OF P.C.F. SELF-SIMILAR NETWORKS." Fractals 28, no. 06 (September 2020): 2050098. http://dx.doi.org/10.1142/s0218348x2050098x.

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In this paper, we discuss a family of p.c.f. self-similar fractal networks which have reflection transformations. We obtain the average geodesic distance on the corresponding fractal in terms of finite pattern of integrals. With these results, we also obtain the asymptotic formula for average distances of the skeleton networks.

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Gaura, Jan, and Eduard Sojka. "Resistance-Geodesic Distance and Its Use in Image Segmentation." International Journal on Artificial Intelligence Tools 25, no. 05 (September 15, 2016): 1640002. http://dx.doi.org/10.1142/s0218213016400029.

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Measuring the distance is an important task in many clustering and image-segmentation algorithms. The value of the distance decides whether two image points belong to a single or, respectively, to two different image segments. The Euclidean distance is used quite often. In more complicated cases, measuring the distances along the surface that is defined by the image function may be more appropriate. The geodesic distance, i.e. the shortest path in the corresponding graph, has become popular in this context. The problem is that it is determined on the basis of only one path that can be viewed as infinitely thin and that can arise accidentally as a result of imperfections in the image. Considering the k shortest paths can be regarded as an effort towards the measurement of the distance that is more reliable. The drawback remains that measuring the distance along several paths is burdened with the same problems as the original geodesic distance. Therefore, it does not guarantee significantly better results. In addition to this, the approach is computationally demanding. This paper introduces the resistance-geodesic distance with the goal to reduce the possibility of using a false accidental path for measurement. The approach can be briefly characterised in such a way that the path of a certain chosen width is sought for, which is in contrast to the geodesic distance. Firstly, the effective conductance is computed for each pair of the neighbouring nodes to determine the local width of the path that could possibly run through the arc connecting them. The width computed in this way is then used for determining the costs of arcs; the arcs whose use would lead to a small width of the final path are penalised. The usual methods for computing the shortest path in a graph are then used to compute the final distances. The corresponding theory and the experimental results are presented in this paper.

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Chen, Da, Jean-Marie Mirebeau, and Laurent D. Cohen. "Vessel tree extraction using radius-lifted keypoints searching scheme and anisotropic fast marching method." Journal of Algorithms & Computational Technology 10, no. 4 (July 7, 2016): 224–34. http://dx.doi.org/10.1177/1748301816656289.

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Geodesic methods have been widely applied to image analysis. They are particularly efficient to extract a tubular structure, such as a blood vessel, given its two endpoints in a 2D or 3D medical image. We address here a more difficult problem: the extraction of a full vessel tree structure given a single initial root point, by growing a collection of keypoints or new initial source points, connected by minimal geodesic paths. In this article, those keypoints are iteratively added, using a new detection criteria, which utilize the weighted geodesic distances with respect to a radius-lifted Riemannian metric, the standard Euclidean curve length and a path score. Two main weaknesses of classical keypoints searching approach are that the weighted geodesic distance and the Euclidean path length do not take into account the orientation of the tubular structure or object boundaries, due to the use of an isotropic geodesic Riemannian metric, and suffer from a leakage problem. In contrast, we use an anisotropic geodesic Riemannian metric, and develop new criteria for selecting keypoints based on the path score and automatically stopping the tree growth. Experimental results demonstrate that our method can obtain the expected results, which can extract vessel structures at a finer scale, with increased accuracy.

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Qin, Xianxiang, Yanning Zhang, Ying Li, Yinglei Cheng, Wangsheng Yu, Peng Wang, and Huanxin Zou. "Distance Measures of Polarimetric SAR Image Data: A Survey." Remote Sensing 14, no. 22 (November 19, 2022): 5873. http://dx.doi.org/10.3390/rs14225873.

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Distance measure plays a critical role in various applications of polarimetric synthetic aperture radar (PolSAR) image data. In recent decades, plenty of distance measures have been developed for PolSAR image data from different perspectives, which, however, have not been well analyzed and summarized. In order to make better use of these distance measures in algorithm design, this paper provides a systematic survey of them and analyzes their relations in detail. We divide these distance measures into five main categories (i.e., the norm distances, geodesic distances, maximum likelihood (ML) distances, generalized likelihood ratio test (GLRT) distances, stochastics distances) and two other categories (i.e., the inter-patch distances and those based on metric learning). Furthermore, we analyze the relations between different distance measures and visualize them with graphs to make them clearer. Moreover, some properties of the main distance measures are discussed, and some advice for choosing distances in algorithm design is also provided. This survey can serve as a reference for researchers in PolSAR image processing, analysis, and related fields.

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Kaya , Abdil, Brecht De Beelde, Wout Joseph, Maarten Weyn, and Rafael Berkvens. "Geodesic Path Model for Indoor Propagation Loss Prediction of Narrowband Channels." Sensors 22, no. 13 (June 29, 2022): 4903. http://dx.doi.org/10.3390/s22134903.

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Indoor path loss models characterize the attenuation of signals between a transmitting and receiving antenna for a certain frequency and type of environment. Their use ranges from network coverage planning to joint communication and sensing applications such as localization and crowd counting. The need for this proposed geodesic path model comes forth from attempts at path loss-based localization on ships, for which the traditional models do not yield satisfactory path loss predictions. In this work, we present a novel pathfinding-based path loss model, requiring only a simple binary floor map and transmitter locations as input. The approximated propagation path is determined using geodesics, which are constrained shortest distances within path-connected spaces. However, finding geodesic paths from one distinct path-connected space to another is done through a systematic process of choosing space connector points and concatenating parts of the geodesic path. We developed an accompanying tool and present its algorithm which automatically extracts model parameters such as the number of wall crossings on the direct path as well as on the geodesic path, path distance, and direction changes on the corners along the propagation path. Moreover, we validate our model against path loss measurements conducted in two distinct indoor environments using DASH-7 sensor networks operating at 868 MHz. The results are then compared to traditional floor-map-based models. Mean absolute errors as low as 4.79 dB and a standard deviation of the model error of 3.63 dB is achieved in a ship environment, almost half the values of the next best traditional model. Improvements in an office environment are more modest with a mean absolute error of 6.16 dB and a standard deviation of 4.55 dB.

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32

Cabello, Sergio. "Computing the Inverse Geodesic Length in Planar Graphs and Graphs of Bounded Treewidth." ACM Transactions on Algorithms 18, no. 2 (April 30, 2022): 1–26. http://dx.doi.org/10.1145/3501303.

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The inverse geodesic length of a graph G is the sum of the inverse of the distances between all pairs of distinct vertices of G . In some domains, it is known as the Harary index or the global efficiency of the graph. We show that, if G is planar and has n vertices, then the inverse geodesic length of G can be computed in roughly O ( n 9/5 ) time. We also show that, if G has n vertices and treewidth at most k , then the inverse geodesic length of G can be computed in O ( n log O ( k ) n ) time. In both cases, we use techniques developed for computing the sum of the distances, which does not have “inverse” component, together with batched evaluations of rational functions.

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33

Orsingher, E., and A. De Gregorio. "Random motions at finite velocity in a non-Euclidean space." Advances in Applied Probability 39, no. 2 (June 2007): 588–611. http://dx.doi.org/10.1239/aap/1183667625.

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In this paper telegraph processes on geodesic lines of the Poincaré half-space and Poincaré disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincaré half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section.

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Orsingher, E., and A. De Gregorio. "Random motions at finite velocity in a non-Euclidean space." Advances in Applied Probability 39, no. 02 (June 2007): 588–611. http://dx.doi.org/10.1017/s0001867800001907.

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In this paper telegraph processes on geodesic lines of the Poincaré half-space and Poincaré disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincaré half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section.

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35

Magnani, Valentino, and Daniele Tiberio. "A remark on vanishing geodesic distances in infinite dimensions." Proceedings of the American Mathematical Society 148, no. 8 (March 4, 2020): 3653–56. http://dx.doi.org/10.1090/proc/14986.

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Lu, Zhipeng, and Xianchang Meng. "Geodesic covers and Erdős distinct distances in hyperbolic surfaces." Annales mathématiques Blaise Pascal 30, no. 2 (April 30, 2024): 201–17. http://dx.doi.org/10.5802/ambp.422.

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Mennucci, Andrea C. G. "On Asymmetric Distances." Analysis and Geometry in Metric Spaces 1 (June 11, 2013): 200–231. http://dx.doi.org/10.2478/agms-2013-0004.

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Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

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KARIMOV, R. KH. "GEODESIC ORBITS AND LYAPUNOV EXPONENTS OF FROLOV'S BLACK HOLE." Izvestia Ufimskogo Nauchnogo Tsentra RAN, no. 2 (June 16, 2023): 34–38. http://dx.doi.org/10.31040/2222-8349-2023-0-2-34-38.

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Binary black holes maintain unstable orbits at very close distances. In the simplest case of geodesics around a Schwarzschild black hole, the orbits, although unstable, are regular and depend only on the mass. In more complex cases, geodesics may depend on charge, rotation, and other parameters. When perturbed, unstable orbits can become a source of chaos. All unstable orbits, whether regular or chaotic, can be quantified by their Lyapunov exponents. Exponents are important for observations because the phase of gravitational waves can decohere in Lyapunov time. If the time scale of dissipation due to gravitational waves is shorter than the Lyapunov time, the chaos will be damped and practically unobservable. These two time scales can be compared. Lyapunov exponents should be used with caution for several reasons: they are relative and dependent on the coordinate system used, they vary from orbit to orbit, and finally, they can be deceptively diluted by transitional behavior for orbits that pass in and out of unstable regions. The stability of circular geodesic orbits of Frolov's black hole space-time is studied in this work. The influence of the black hole charge and the scale parameter on the stability of geodesic orbits and the Lyapunov exponent is analyzed. It is shown that the region of stable circular orbits increases with the black hole charge Q and the scale parameter ℓ . The largest region of stable circular orbits of Frolov's black hole is reached at Q = M and ℓ = 0.75M.

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CHEN, JUHUA, and YONGJIU WANG. "TIMELIKE GEODESIC MOTION IN HORAVA–LIFSHITZ SPACE–TIME." International Journal of Modern Physics A 25, no. 07 (March 20, 2010): 1439–48. http://dx.doi.org/10.1142/s0217751x10048962.

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Recently a nonrelativistic renormalizable theory of gravitation has been proposed by P. Horava. When restricted to satisfy the condition of detailed balance, this theory is intimately related to topologically massive gravity in three dimensions, and the geometry of the Cotton tensor. At long distances, this theory is expected to flow to the relativistic value λ = 1, and could therefore serve as a possible candidate for a UV completion of Einstein's general relativity or an infrared modification thereof. In this paper under allowing the lapse function to depend on the spatial coordinates xi as well as t, we obtain the spherically symmetric solutions. And then by analyzing the behavior of the effective potential for the particle, we investigate the timelike geodesic motion of particle in the Horava–Lifshitz space–time. We find that the nonradial particle falls from a finite distance to the center along the timelike geodesics when its energy is in an appropriate range. However, we find that it is complexity for radial particle along the timelike geodesics. There are three different cases due to the energy of radial particle: (i) when the energy of radial particle is higher than a critical value EC, the particle will fall directly from infinity to the singularity; (ii) when the energy of radial particle equals to the critical value EC, the particle orbit at r = rC is unstable, i.e. the particle will escape from r = rC to the infinity or to the singularity, depending on the initial conditions of the particle; (iii) when the energy of radial particle is in a proper range, the particle will rebound to the infinity or plunge to the singularity from a infinite distance, depending on the initial conditions of the particle.

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YE, QIANQIAN, and LIFENG XI. "AVERAGE DISTANCE OF SUBSTITUTION NETWORKS." Fractals 27, no. 06 (September 2019): 1950097. http://dx.doi.org/10.1142/s0218348x1950097x.

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The substitution network is a deterministic model of evolving self-similar networks. For normalized substitution networks, the limit of metric spaces with respect to networks is a self-similar fractal and the limit of average distances on networks is the integral of geodesic distance of the fractal on the self-similar measure. After some technical handles, we establish the finiteness of integrals and obtain a linear equation set to solve the average distance on the fractal.

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Hamenstädt, Ursula. "Time-preserving conjugacies of geodesic flows." Ergodic Theory and Dynamical Systems 12, no. 1 (March 1992): 67–74. http://dx.doi.org/10.1017/s0143385700006581.

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AbstractIn this note we study Borel-probability measures on the unit tangent bundle ofa compact negatively curved manifold M that are invariant under the geodesic flow. We interpret the entropy of such a measure as a Hausdorff dimension with respect to a natural family of distances on the ideal boundary of the universal covering of M. This in term yields necessary and sufficient conditions for the existence of time preserving conjugacies of geodesic flows.

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Gattone, Stefano, Angela De Sanctis, Stéphane Puechmorel, and Florence Nicol. "On the Geodesic Distance in Shapes K-means Clustering." Entropy 20, no. 9 (August 29, 2018): 647. http://dx.doi.org/10.3390/e20090647.

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In this paper, the problem of clustering rotationally invariant shapes is studied and a solution using Information Geometry tools is provided. Landmarks of a complex shape are defined as probability densities in a statistical manifold. Then, in the setting of shapes clustering through a K-means algorithm, the discriminative power of two different shapes distances are evaluated. The first, derived from Fisher–Rao metric, is related with the minimization of information in the Fisher sense and the other is derived from the Wasserstein distance which measures the minimal transportation cost. A modification of the K-means algorithm is also proposed which allows the variances to vary not only among the landmarks but also among the clusters.

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Wang, Ziyun, Eric A. Mitchell, Volkan Isler, and Daniel D. Lee. "Geodesic-HOF: 3D Reconstruction Without Cutting Corners." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 4 (May 18, 2021): 2844–51. http://dx.doi.org/10.1609/aaai.v35i4.16390.

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Single-view 3D object reconstruction is a challenging fundamental problem in machine perception, largely due to the morphological diversity of objects in the natural world. In particular, high curvature regions are not always represented accurately by methods trained with common set-based loss functions such as Chamfer Distance, resulting in reconstructions short-circuiting the surface or "cutting corners." To address this issue, we propose an approach to 3D reconstruction that embeds points on the surface of an object into a higher-dimensional space that captures both the original 3D surface as well as geodesic distances between points on the surface of the object. The precise specification of these additional "lifted" coordinates ultimately yields useful surface information without requiring excessive additional computation during either training or testing, in comparison with existing approaches. Our experiments show that taking advantage of these learned lifted coordinates yields better performance for estimating surface normals and generating surfaces than using point cloud reconstructions alone. Further, we find that this learned geodesic embedding space provides useful information for applications such as unsupervised object decomposition.

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Walwyn, P. R. "The Great Ellipse Solution for Distances and Headings to Steer between Waypoints." Journal of Navigation 52, no. 3 (September 1999): 421–24. http://dx.doi.org/10.1017/s0373463399008516.

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The normal Great Circle method of computing the shortest distance between two positions on the Earth – e.g. from an aircraft's present position (PP) to a waypoint (WP) – is not accurate enough to meet present-day requirements for aircraft Nav–Attack systems.On the surface of an Ellipsoid (or Spheroid), the true ‘shortest distance’ is along a geodesic curve between the two points, but the computation of this curve is complex, and as shown by R. Williams at Reference, the difference between the geodesic and Great Ellipse distances between two points is negligible (<0·01 nm).The Great Ellipse through two points on a spheroid is defined as the ellipse that passes through the two points and the centre of the spheroid; it therefore has a major axis equal to the Earth's, and a minor axis that is between the Earth's major axis (for two points on the Equator) and minor axis (for two points on the same, or diametrically opposite, longitudes). Thus the problem of deciding on which Great Ellipse the two points lie is equivalent to determining the magnitude of the minor axis β of the ellipse on which they both lie.

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Karbauskaitė, Rasa, and Gintautas Dzemyda. "Geodesic distances in the intrinsic dimensionality estimation using packing numbers." Nonlinear Analysis: Modelling and Control 19, no. 4 (December 10, 2014): 578–91. http://dx.doi.org/10.15388/na.2014.4.4.

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Karbauskaitė, Rasa, Gintautas Dzemyda, and Edmundas Mazėtis. "Geodesic distances in the maximum likelihood estimator of intrinsic dimensionality." Nonlinear Analysis: Modelling and Control 16, no. 4 (December 7, 2011): 387–402. http://dx.doi.org/10.15388/na.16.4.14084.

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While analyzing multidimensional data, we often have to reduce their dimensionality so that to preserve as much information on the analyzed data set as possible. To this end, it is reasonable to find out the intrinsic dimensionality of the data. In this paper, two techniques for the intrinsic dimensionality are analyzed and compared, i.e., the maximum likelihood estimator (MLE) and ISOMAP method. We also propose the way how to get good estimates of the intrinsic dimensionality by the MLE method.

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Campen, Marcel, and Leif Kobbelt. "Walking On Broken Mesh: Defect-Tolerant Geodesic Distances and Parameterizations." Computer Graphics Forum 30, no. 2 (April 2011): 623–32. http://dx.doi.org/10.1111/j.1467-8659.2011.01896.x.

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Yang, Xiangli, Wen Yang, Hui Song, and Pingping Huang. "Polarimetric SAR Image Classification Using Geodesic Distances and Composite Kernels." IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing 11, no. 5 (May 2018): 1606–14. http://dx.doi.org/10.1109/jstars.2018.2802045.

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Battagliero, S., G. Puglia, S. Vicario, F. Rubino, G. Scioscia, and P. Leo. "An Efficient Algorithm for Approximating Geodesic Distances in Tree Space." IEEE/ACM Transactions on Computational Biology and Bioinformatics 8, no. 5 (September 2011): 1196–207. http://dx.doi.org/10.1109/tcbb.2010.121.

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Owen, Megan, and J. Scott Provan. "A Fast Algorithm for Computing Geodesic Distances in Tree Space." IEEE/ACM Transactions on Computational Biology and Bioinformatics 8, no. 1 (January 2011): 2–13. http://dx.doi.org/10.1109/tcbb.2010.3.

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