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Journal articles on the topic 'Geodesical projection'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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1

Bejan, Cornelia-Livia, and Simona-Luiza Druţǎ-Romaniuc. "F-geodesics on manifolds." Filomat 29, no. 10 (2015): 2367–79. http://dx.doi.org/10.2298/fil1510367b.

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The notion of F-geodesic, which is slightly different from that of F-planar curve (see [13], [17], and [18]), generalizes the magnetic curves, and implicitly the geodesics, by using any (1,1)-tensor field on the manifold (in particular the electro-magnetic field or the Lorentz force). We give several examples of F-geodesics and the characterizations of the F-geodesics w.r.t. Vranceanu connections on foliated manifolds and adapted connections on almost contact manifolds. We generalize the classical projective transformation, holomorphic-projective transformation and C-projective transformation, by considering a pair of symmetric connections which have the same F-geodesics. We deal with the transformations between such two connections, namely F-planar diffeomorphisms ([18]). We obtain a Weyl type tensor field, invariant under any F-planar diffeomorphism, on a 1-codimensional foliation.

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KIM, BYUNG HAK, IN-BAE KIM, and SADAHIRO MAEDA. "CHARACTERIZATIONS OF BERGER SPHERES FROM THE VIEWPOINT OF SUBMANIFOLD THEORY." Glasgow Mathematical Journal 62, no. 1 (March 1, 2019): 137–45. http://dx.doi.org/10.1017/s0017089519000016.

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AbstractIn this paper, Berger spheres are regarded as geodesic spheres with sufficiently big radii in a complex projective space. We characterize such real hypersurfaces by investigating their geodesics and contact structures from the viewpoint of submanifold theory.

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3

Le Donne, Enrico, and Roger Züst. "Space of signatures as inverse limits of Carnot groups." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 37. http://dx.doi.org/10.1051/cocv/2021040.

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We formalize the notion of limit of an inverse system of metric spaces with 1-Lipschitz projections having unbounded fibers. The construction is applied to the sequence of free Carnot groups of fixed rank n and increasing step. In this case, the limit space is in correspondence with the space of signatures of rectifiable paths in ℝn, as introduced by Chen. Hambly-Lyons’s result on the uniqueness of signature implies that this space is a geodesic metric tree. As a particular consequence we deduce that every path in ℝn can be approximated by projections of some geodesics in some Carnot group of rank n, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step.

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4

Leininger, Christopher, Anna Lenzhen, and Kasra Rafi. "Limit sets of Teichmüller geodesics with minimal non-uniquely ergodic vertical foliation." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 737 (April 1, 2018): 1–32. http://dx.doi.org/10.1515/crelle-2015-0040.

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AbstractWe describe a method for constructing Teichmüller geodesics where the vertical foliation ν is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the Teichmüller geodesic. The construction depends on various parameters, and we show that one can adjust the parameters to ensure that the set of accumulation points of such a geodesic in the Thurston boundary is exactly the projective 1-simplex of all projective measured foliations that are topologically equivalent to ν. With further adjustment of the parameters, one can further assume that the transverse measure is an ergodic measure on the non-uniquely ergodic foliation ν.

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5

BUCATARU, IOAN, and ZOLTÁN MUZSNAY. "PROJECTIVE AND FINSLER METRIZABILITY: PARAMETERIZATION-RIGIDITY OF THE GEODESICS." International Journal of Mathematics 23, no. 09 (July 31, 2012): 1250099. http://dx.doi.org/10.1142/s0129167x12500991.

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In this work we show that for the geodesic spray S of a Finsler function F, the most natural projective deformation [Formula: see text] leads to a non-Finsler metrizable spray, for almost every value of λ ∈ ℝ. This result shows how rigid is the metrizablility property with respect to certain reparameterizations of the geodesics. As a consequence, we obtain that the projective class of an arbitrary spray contains infinitely many sprays that are not Finsler metrizable.

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6

Lenells, Jonatan. "Spheres, Kähler geometry and the Hunter–Saxton system." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2154 (June 8, 2013): 20120726. http://dx.doi.org/10.1098/rspa.2012.0726.

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Many important equations of mathematical physics arise geometrically as geodesic equations on Lie groups. In this paper, we study an example of a geodesic equation, the two-component Hunter–Saxton (2HS) system, which displays a number of unique geometric features. We show that 2HS describes the geodesic flow on a manifold, which is isometric to a subset of a sphere. Since the geodesics on a sphere are simply the great circles, this immediately yields explicit formulae for the solutions of 2HS. We also show that when restricted to functions of zero mean, 2HS reduces to the geodesic equation on an infinite-dimensional manifold, which admits a Kähler structure. We demonstrate that this manifold is in fact isometric to a subset of complex projective space, and that the above constructions provide an example of an infinite-dimensional Hopf fibration.

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7

ANDRUCHOW, ESTEBAN, and LÁZARO RECHT. "GRASSMANNIANS OF A FINITE ALGEBRA IN THE STRONG OPERATOR TOPOLOGY." International Journal of Mathematics 17, no. 04 (April 2006): 477–91. http://dx.doi.org/10.1142/s0129167x06003552.

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If [Formula: see text] is a type II1 von Neumann algebra with a faithful trace τ, we consider the set [Formula: see text] of self-adjoint projections of [Formula: see text] as a subset of the Hilbert space [Formula: see text]. We prove that though it is not a differentiable submanifold, the geodesics of the natural Levi–Civita connection given by the trace have minimal length. More precisely: the curves of the form γ(t) = eitxpe-itx with x* = x, pxp = (1 - p)x(1 - p) = 0 have minimal length when measured in the Hilbert space norm of [Formula: see text], provided that the operator norm ‖x‖ is less or equal than π/2. Moreover, any two projections which are unitary equivalent are joined by at least one such minimal geodesic, and only unitary equivalent projections can be joined by a smooth curve. Finally, we prove that these geodesics have also minimal length if one measures them with the Schatten k-norms of τ, ‖x‖k = τ((x* x)k/2)1/k, for all k ∈ ℝ, k ≥ 0. We also characterize curves of unitaries which have minimal length with these k-norms.

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8

Pędzich, Paweł, and Marta Kuźma. "Application of methods for area calculation of geodesic polygons on Polish administrative units." Geodesy and Cartography 61, no. 2 (November 1, 2012): 105–15. http://dx.doi.org/10.2478/v10277-012-0025-6.

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Abstract The paper presents methods of area calculation, which may be applied for big geodesic polygons on the ellipsoid. Proposal developed by the authors of this paper is discussed. The proposed methods are compared with other, alternative methods of area calculation of such polygons. Test calculations are performed for administrative units in Poland. The obtained results are also compared with areas of those units registered in statistical annals. Utilisation of the equal-area map projections of the ellipsoid onto a plane seems to be the best solution for the discussed task. In the case of small distances between points we may expect accurate results of calculations, since the area size is influenced by the projection reductions only, which are small in such cases. In some cases their influence on results of calculations may be neglected. Then, only re-calculation of co-ordinates from the GRS80 ellipsoid to the cartographic, equal-area projection is required.

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9

Williams, Roy. "Gnomonic Projection of the Surface of an Ellipsoid." Journal of Navigation 50, no. 2 (May 1997): 314–20. http://dx.doi.org/10.1017/s0373463300023936.

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When a surface is mapped onto a plane so that the image of a geodesic arc is a straight line on the plane then the mapping is known as a geodesic mapping. It is only possible to perform a geodesic mapping of a surface onto a plane when the surface has constant normal curvature. The normal curvature of a sphere of radius r at all points on the surface is I/r hence it is possible to map the surface of a sphere onto a plane using a geodesic mapping. The geodesic mapping of the surface of a sphere onto a plane is achieved by a gnomonic projection which is the projection of the surface of the sphere from its centre onto a tangent plane. There is no geodesic mapping of the ellipsoid of revolution or the spheroid onto a plane because the ellipsoid of revolution or the spheroid are not surfaces whose curvature is constant at all points. We can, however, still construct a projection of the surface of the ellipsoid from the centre of the body onto a tangent plane and we call this projection a gnomonic projection also.

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10

Podestà, Fabio. "Projective submersions." Bulletin of the Australian Mathematical Society 43, no. 2 (April 1991): 251–56. http://dx.doi.org/10.1017/s0004972700029014.

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We consider C∞ manifolds endowed with torsionfree affine connections and C∞ projective submersions between them which, by definition, map geodesics into geodesics up to parametrisation. After giving a differential characterisation of these mappings, we deal with the case when one of the given connections is projectively flat or satisfies certain conditions concerning its Ricci tensor; under these hypotheses we prove that the projective submersion is actually a covering.

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11

Crider, John E. "A Geodesic Map Projection for Quadrilaterals." Cartography and Geographic Information Science 36, no. 2 (January 2009): 131–47. http://dx.doi.org/10.1559/152304009788188781.

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12

Martínez-Llario, José Carlos, Sergio Baselga, and Eloína Coll. "Accurate Algorithms for Spatial Operations on the Spheroid in a Spatial Database Management System." Applied Sciences 11, no. 11 (May 31, 2021): 5129. http://dx.doi.org/10.3390/app11115129.

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Some of the most powerful spatial analysis software solutions (Oracle, Google Earth Engine, PostgreSQL + PostGIS, etc.) are currently performing geometric calculations directly on the ellipsoid (a quadratic surface that models the earth shape), with a double purpose: to attain a high degree of accuracy and to allow the full management of large areas of territory (countries or even continents). It is well known that both objectives are impossible to achieve by means of the traditional approach using local mathematical projections and Cartesian coordinates. This paper demonstrates in a quantitative methodological way that most of the spatial analysis software products make important deviations in calculations regarding to geodesics, being the users unaware of the magnitude of these inaccuracies, which can easily reach meters depending on the distance. This is due to the use of ellipsoid calculations in an approximate way (e.g., using a sphere instead of an ellipsoid). This paper presents the implementation of two algorithms that solve with high accuracy (less than 100 nm) and efficiently (few iterations) two basic geometric calculations on the ellipsoid that are essential to build more complex spatial operators: the intersection of two geodesics and the minimum distance from a point to a geodesic.

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13

Maeda, Sadahiro, and Koichi Ogiue. "Characterizations of geodesic hyperspheres in a complex projective space by observing the extrinsic shape of geodesics." Mathematische Zeitschrift 225, no. 4 (August 7, 1997): 537–42. http://dx.doi.org/10.1007/pl00004625.

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14

Guo, Hui. "Geodesic uniqueness and derivatives of Bers projection." Bulletin of the Australian Mathematical Society 67, no. 1 (February 2003): 145–62. http://dx.doi.org/10.1017/s0004972700033608.

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15

Wang, Chengen, and Qiang Liu. "Projection and Geodesic-Based Pipe Routing Algorithm." IEEE Transactions on Automation Science and Engineering 8, no. 3 (July 2011): 641–45. http://dx.doi.org/10.1109/tase.2010.2099219.

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16

Matveev, S. I., and A. S. Matveev. "On the establishment of an unified global rectangular coordinates for the whole of Russia." Geodesy and Cartography 919, no. 1 (February 20, 2017): 52–54. http://dx.doi.org/10.22389/0016-7126-2017-919-1-52-54.

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This article deals with the problem of creating unitary global space orthogonal geocentric coordinatesystem throughout Russia on the base of a geoid. This problem arose on Resolution of The Russian Federation Government. The point is that geodesic coordinates B, L and H are acceptable for moving and navigation of transport, but orthogonal coordinates X and Y are in the equatorial plane and have nothing to do with designing and construction. Adaptive systems unite global orthogonal coordinates. That is why the authors set the goal of using polyhedral projections. All maps compiled and used in Russian cartographic Gauss and Crueger projection could be attributed to that. For many border squares errors will be very little. The authors used very little borders 20 by 20 km and the known topo-centric methods. As a result they create an adaptive system of unitary global orthogonal coordinates.

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17

Guillemin, V., A. Uribe, and Z. Wang. "Geodesics on weighted projective spaces." Annals of Global Analysis and Geometry 36, no. 2 (March 19, 2009): 205–20. http://dx.doi.org/10.1007/s10455-009-9159-7.

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18

POPOV, G., and P. TOPALOV. "Discrete analog of the projective equivalence and integrable billiard tables." Ergodic Theory and Dynamical Systems 28, no. 5 (October 2008): 1657–84. http://dx.doi.org/10.1017/s014338570700096x.

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AbstractA class of discrete dynamical systems called projectively (or geodesically) equivalent Lagrangian systems is defined. We prove that these systems admit families of integrals. In the case of geodesically equivalent billiard tables, these integrals are pairwise commuting. We describe a family of geodesically equivalent billiard tables on surfaces of constant curvature. This is a special case of the so-called ‘Liouville billiard tables’.

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19

Simić, Slobodan N. "On the sub-Riemannian geometry of contact Anosov flows." Journal of Topology and Analysis 08, no. 01 (February 23, 2016): 187–205. http://dx.doi.org/10.1142/s1793525316500072.

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We investigate certain natural connections between sub-Riemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a sub-Riemannian geodesic shared between its projections onto the integrable summands? We show that if the horizontal distribution is the sum of the strong stable and strong unstable distributions of a special type of a contact Anosov flow in three dimensions, then for any short enough sub-Riemannian geodesic connecting points on the same orbit of the Anosov flow, the energy of the geodesic is shared equally between its projections onto the stable and unstable bundles. The proof relies on a connection between the geodesic equations and the harmonic oscillator equation, and its explicit solution by the Jacobi elliptic functions. Using a different idea, we prove an analogous result in higher dimensions for the geodesic flow of a closed Riemannian manifold of constant negative curvature.

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20

Andruchow, Esteban. "Geodesics of projections in von Neumann algebras." Proceedings of the American Mathematical Society 149, no. 10 (July 28, 2021): 4501–13. http://dx.doi.org/10.1090/proc/15568.

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Let A {\mathcal {A}} be a von Neumann algebra and P A {\mathcal {P}}_{\mathcal {A}} the manifold of projections in A {\mathcal {A}} . There is a natural linear connection in P A {\mathcal {P}}_{\mathcal {A}} , which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of C n \mathbb {C}^n . In this paper we show that two projections p , q p,q can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of A {\mathcal {A}} ), if and only if p ∧ q ⊥ ∼ p ⊥ ∧ q , \begin{equation*} p\wedge q^\perp \sim p^\perp \wedge q, \end{equation*} where ∼ \sim stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal geodesic is unique if and only if p ∧ q ⊥ = p ⊥ ∧ q = 0 p\wedge q^\perp = p^\perp \wedge q=0 . If A {\mathcal {A}} is a finite factor, any pair of projections in the same connected component of P A {\mathcal {P}}_{\mathcal {A}} (i.e., with the same trace) can be joined by a minimal geodesic. We explore certain relations with Jones’ index theory for subfactors. For instance, it is shown that if N ⊂ M {\mathcal {N}}\subset {\mathcal {M}} are II 1 _1 factors with finite index [ M : N ] = t − 1 [{\mathcal {M}}:{\mathcal {N}}]={\mathbf {t}}^{-1} , then the geodesic distance d ( e N , e M ) d(e_{\mathcal {N}},e_{\mathcal {M}}) between the induced projections e N e_{\mathcal {N}} and e M e_{\mathcal {M}} is d ( e N , e M ) = arccos ⁡ ( t 1 / 2 ) d(e_{\mathcal {N}},e_{\mathcal {M}})=\arccos ({\mathbf {t}}^{1/2}) .

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21

Savas, Murat, Baki Karliga, and Atakan T. Yakut. "Orthogonal Projections Based on Hyperbolic and Sphericaln-Simplex." Advances in Mathematical Physics 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/808250.

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Orthogonal projection along a geodesic to the chosenk-plane is introduced using edge and Gram matrix of ann-simplex in hyperbolic or sphericaln-space. The distance from a point tok-plane is obtained by the orthogonal projection. It is also given the perpendicular foot from a point tok-plane of hyperbolic and sphericaln-space.

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22

Bakhoum, Ezzat G. "Gaussian Curvature in Propagation Problems in Physics and Engineering." Mathematical Problems in Engineering 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/371890.

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The computation of the Gaussian curvature of a surface is a requirement in many propagation problems in physics and engineering. A formula is developed for the calculation of the Gaussian curvature by knowledge of two close geodesics on the surface, or alternatively from the projection (i.e., image) of such geodesics. The formula will be very useful for problems in general relativity, civil engineering, and robotic navigation.

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23

Kovarik, Zdislav V., and Nagwa Sherif. "Geodesics and near-geodesics in the manifolds of projector frames." Linear Algebra and its Applications 99 (February 1988): 259–77. http://dx.doi.org/10.1016/0024-3795(88)90136-x.

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24

Pries, Christian. "Geodesics Closed On The Projective Plane." Geometric and Functional Analysis 18, no. 5 (December 22, 2008): 1774–85. http://dx.doi.org/10.1007/s00039-008-0682-7.

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25

Fernández-Serrano, Martino Peña, and José Calvo López. "Projecting Stars, Triangles and Concrete." Architectura 47, no. 1-2 (July 24, 2019): 92–114. http://dx.doi.org/10.1515/atc-2017-0006.

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AbstractSometimes scientific-technical objects can be given an extended meaning as cultural icons and be received in art and architecture. To this end, the object must be detached from its original context and viewed from different, new perspectives.In 1922 Walter Bauersfeld constructed one of the first geodesic domes for testing projection devices in Jena. Walter Gropius and Lázló Moholy-Nagy were among the first to visit the Jena Planetarium; Moholy-Nagy received the dome in his book ›Von Material zu Architektur‹. Richard Buckminster Fuller further developed Bauersfeld’s concept from the 1940s and patented the construction principle of a geodesic dome under the name ›Building Construction‹ in 1954. His patent bears resemblances to the Bauersfeld Planetarium in Jena, which can be demonstrated by manuscripts by Bauersfeld from the Zeiss Archive in Jena. Fuller, on the other hand, also used the geodesic dome to explain his theory as Synergetic. The article traces the transformation of the technical object conceived by Bauersfeld via Moholy-Nagy and Fuller into a cultural icon of the 20th century.

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26

Gudmundsson, Sigmundur. "Harmonic Morphisms as Sphere Bundles Over Compact Riemann Surfaces." International Journal of Mathematics 08, no. 07 (November 1997): 935–42. http://dx.doi.org/10.1142/s0129167x97000445.

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27

Gotoh, Tohru. "Geodesic hyperspheres in complex projective space." Tsukuba Journal of Mathematics 18, no. 1 (June 1994): 207–15. http://dx.doi.org/10.21099/tkbjm/1496162466.

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28

Adachi, Toshiaki. "Distribution of closed geodesics with a preassigned homology class in a negatively curved manifold." Nagoya Mathematical Journal 110 (June 1988): 1–14. http://dx.doi.org/10.1017/s0027763000002865.

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Let M be a compact Riemannian manifold whose geodesic flow φi : UM→UM on the unit tangent bundle is of Anosov type. In this paper we count the number of φi-closed orbits and study the distribution of prime closed geodesies in a given homology class in H1(M, Z). Here a prime closed geodesic means an (oriented) image of a φi-closed orbit by the projection p : UM → M.

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29

Andruchow, Esteban. "Pairs of Projections: Geodesics, Fredholm and Compact Pairs." Complex Analysis and Operator Theory 8, no. 7 (August 15, 2013): 1435–53. http://dx.doi.org/10.1007/s11785-013-0327-1.

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30

Fiori, Simone. "Geodesic-based and projection-based neural blind deconvolution algorithms." Signal Processing 88, no. 3 (March 2008): 521–38. http://dx.doi.org/10.1016/j.sigpro.2007.08.014.

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31

BONATTI, CHRISTIAN, XAVIER GÓMEZ-MONT, and MATILDE MARTÍNEZ. "Foliated hyperbolicity and foliations with hyperbolic leaves." Ergodic Theory and Dynamical Systems 40, no. 4 (September 17, 2018): 881–903. http://dx.doi.org/10.1017/etds.2018.61.

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Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behavior of almost every $X$-orbit in every leaf, which we call Gibbs $u$-states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using the foliated hyperbolic vector field on the unit tangent bundle to the foliation generating the leaf geodesics. When the Lyapunov exponents of such ergodic Gibbs $u$-states are negative, it is an SRB measure (having a positive Lebesgue basin of attraction). When the foliation is by hyperbolic leaves, this class of probabilities coincide with the classical harmonic measures introduced by Garnett. Furthermore, if the foliation is transversally conformal and does not admit a transverse invariant measure we show that there are finitely many ergodic Gibbs $u$-states, each supported in one minimal set of the foliation, each having negative Lyapunov exponents, and the union of their basins of attraction has full Lebesgue measure. The leaf geodesics emanating from a point have a proportion whose asymptotic statistics are described by each of these ergodic Gibbs $u$-states, giving rise to continuous visibility functions of the attractors. Reversing time, by considering $-X$, we obtain the existence of the same number of repellers of the foliated geodesic flow having the same harmonic measures as projections to $M$. In the case of only one attractor, we obtain a north to south pole dynamics.

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32

Lin, Samuel, and Benjamin Schmidt. "Real projective spaces with all geodesics closed." Geometric and Functional Analysis 27, no. 3 (March 20, 2017): 631–36. http://dx.doi.org/10.1007/s00039-017-0407-x.

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33

Minguzzi, E. "Proper time and conformal problem in Kaluza–Klein theory." International Journal of Geometric Methods in Modern Physics 12, no. 05 (May 2015): 1550063. http://dx.doi.org/10.1142/s0219887815500632.

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In the traditional Kaluza–Klein theory, the cylinder condition and the constancy of the extra-dimensional radius (scalar field) imply that time-like geodesics on the five-dimensional bundle project to solutions of the Lorentz force equation on spacetime. This property is lost for nonconstant scalar fields, in fact there appears new terms that have been interpreted mainly as new forces or as due to a variable inertial mass and/or charge. Here we prove that the additional terms can be removed if we assume that charged particles are coupled with the same spacetime conformal structure of neutral particles but through a different conformal factor. As a consequence, in Kaluza–Klein theory the proper time of the charged particle might depend on the charge-to-mass ratio and the scalar field. Then we show that the compatibility between the equation of the projected geodesic and the classical limit of the Klein–Gordon equation fixes unambiguously the conformal factor of the coupling metric solving the conformal ambiguity problem of Kaluza–Klein theories. We confirm this result by explicitly constructing the projection of the Klein–Gordon equation and by showing that each Fourier mode, even for a variable scalar field, satisfies the Klein–Gordon equation on the base.

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34

Lange, Christian. "On metrics on 2-orbifolds all of whose geodesics are closed." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 758 (January 1, 2020): 67–94. http://dx.doi.org/10.1515/crelle-2017-0050.

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AbstractWe show that the periods and the topology of the space of closed geodesics on a Riemannian 2-orbifold all of whose geodesics are closed depend, up to scaling, only on the orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length, without referring to the existence of simple geodesics. We partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the projective plane based on a systolic inequality due to Pu.

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35

MAEDA, SADAHIRO, and TOSHIAKI ADACHI. "CHARACTERIZATIONS OF HYPERSURFACES OF TYPE A2 IN A COMPLEX PROJECTIVE SPACE." Bulletin of the Australian Mathematical Society 77, no. 1 (February 2008): 1–8. http://dx.doi.org/10.1017/s0004972708000014.

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36

Zlatanović, Milan Lj. "New projective tensors for equitorsion geodesic mappings." Applied Mathematics Letters 25, no. 5 (May 2012): 890–97. http://dx.doi.org/10.1016/j.aml.2011.10.045.

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37

Sabykanov, Almazbek, Josef Mikes, and Patrik Peska. "Recurrent equiaffine projective Euclidean spaces." Filomat 33, no. 4 (2019): 1053–58. http://dx.doi.org/10.2298/fil1904053s.

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In this paper, we study n-dimensional recurrent equiaffine projective Euclidean manifolds, i.e. manifolds with absolute recurrent curvature tensor, which admit geodesic mappings onto Euclidean space, and they are equiaffine (where was obtained the symmetric Ricci tensor). We obtained main conditions of recurrent projective Euclidean spaces and constructed their examples.

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Hovila, Risto, Esa Järvenpää, Maarit Järvenpää, and François Ledrappier. "Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces." Geometriae Dedicata 161, no. 1 (January 14, 2012): 51–61. http://dx.doi.org/10.1007/s10711-012-9693-5.

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39

Montagne, Nicolas, Cyril Douthe, Xavier Tellier, Corentin Fivet, and Olivier Baverel. "Voss Surfaces: A Design Space for Geodesic Gridshells." Journal of the International Association for Shell and Spatial Structures 61, no. 4 (December 1, 2020): 255–63. http://dx.doi.org/10.20898/j.iass.2020.008.

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The design of envelopes with complex geometries often leads to construction challenges. To overcome these difficulties, resorting to discrete differential geometry proved successful by establishing close links between mesh properties and the existence of good fabrication, assembling and mechanical properties. In this paper, the design of a special family of structures, called geodesic shells, is addressed using Voss nets, a family of discrete surfaces. The use of discrete Voss surfaces ensures that the structure can be built from simply connected, initially straight laths, and covered with flat panels. These advantageous constructive properties arise from the existence of a conjugate network of geodesic curves on the underlying smooth surface. Here, a review of Voss nets is presented and particular attention is given to the projection of normal vectors on the unit sphere. This projection, called Gauss map, creates a dual net which unveils the remarkable characteristics of Voss nets. Then, based on the previous study, two generation methods are introduced. One enables the exploration and the deformation of Voss nets while the second provides a more direct computational technique. The application of theses methodologies is discussed alongside formal examples.

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van den Boogaart, K. Gerald, and Helmut Schaeben. "Exploratory orientation data analysis with ω sections." Journal of Applied Crystallography 37, no. 5 (September 11, 2004): 683–97. http://dx.doi.org/10.1107/s0021889804011446.

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Since the domain of crystallographic orientations is three-dimensional and spherical, insightful visualization of them or visualization of related probability density functions requires (i) exploitation of the effect of a given orientation on the crystallographic axes, (ii) consideration of spherical means of the orientation probability density function, in particular with respect to one-dimensional totally geodesic submanifolds, and (iii) application of projections from the two-dimensional unit sphere S^2 \subset I\!R^3 onto the unit disk D \subset I\!R^2. The familiar crystallographic `pole figures' are actually mean values of the spherical Radon {\cal R}_1 transform. The mathematical Radon {\cal R}_1 transform associates a real-valued functionfdefined on a sphere with its mean values {\cal R}_{1}f along one-dimensional circles with centre {\cal O}, the origin of the coordinate system, and spanned by two unit vectors. The family of views suggested here defines ω sections in terms of simultaneous orientational relationships of two different crystal axes with two different specimen directions, such that their superposition yields a user-specified pole probability density function. Thus, the spherical averaging and the spherical projection onto the unit disk determine the distortion of the display. Commonly, spherical projections preserving either volume or angle are favoured. This rich family displaysfcompletely,i.e.iffis given or can be determined unambiguously, then it is uniquely represented by several subsets of these views. A computer code enables the user to specify and control interactively the display of linked views, which is comprehensible as the user is in control of the display.

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Nowak, Edward. "Geodesic survey and modernization of a route as the task of optimization." Geodesy and Cartography 63, no. 1 (June 1, 2014): 75–87. http://dx.doi.org/10.2478/geocart-2014-0006.

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Abstract A geodesic survey of an existing route requires one to determine the approximation curve by means of optimization using the total least squares method (TLSM). The objective function of the LSM was found to be a square of the Mahalanobis distance in the adjustment field ν. In approximation tasks, the Mahalanobis distance is the distance from a survey point to the desired curve. In the case of linear regression, this distance is codirectional with a coordinate axis; in orthogonal regression, it is codirectional with the normal line to the curve. Accepting the Mahalanobis distance from the survey point as a quasi-observation allows us to conduct adjustment using a numerically exact parametric procedure. Analysis of the potential application of splines under the NURBS (non-uniform rational B-spline) industrial standard with respect to route approximation has identified two issues: a lack of the value of the localizing parameter for a given survey point and the use of vector parameters that define the shape of the curve. The value of the localizing parameter was determined by projecting the survey point onto the curve. This projection, together with the aforementioned Mahalanobis distance, splits the position vector of the curve into two orthogonal constituents within the local coordinate system of the curve. A similar system corresponds to points that form the control polygonal chain and allows us to find their position with the help of a scalar variable that determines the shape of the curve by moving a knot toward the normal line.

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MATVEEV, VLADIMIR S., and STEFAN ROSEMANN. "TWO REMARKS ON PQε-PROJECTIVITY OF RIEMANNIAN METRICS." Glasgow Mathematical Journal 55, no. 1 (August 2, 2012): 131–38. http://dx.doi.org/10.1017/s0017089512000390.

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AbstractWe show that PQε-projectivity of two Riemannian metrics introduced in [15] (P. J. Topalov, Geodesic compatibility and integrability of geodesic flows, J. Math. Phys.44(2) (2003), 913–929.) implies affine equivalence of the metrics unless ε ∈ {0,−1,−3,−5,−7,. . .}. Moreover, we show that for ε=0, PQε-projectivity implies projective equivalence.

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Sutton, Craig J. "Measures invariant under the geodesic flow and their projections." Proceedings of the American Mathematical Society 131, no. 9 (April 9, 2003): 2933–36. http://dx.doi.org/10.1090/s0002-9939-03-07136-3.

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Paliathanasis, Andronikos. "Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations." Symmetry 13, no. 6 (June 6, 2021): 1018. http://dx.doi.org/10.3390/sym13061018.

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We investigate the relation of the Lie point symmetries for the geodesic equations with the collineations of decomposable spacetimes. We review previous results in the literature on the Lie point symmetries of the geodesic equations and we follow a previous proposed geometric construction approach for the symmetries of differential equations. In this study, we prove that the projective collineations of a n+1-dimensional decomposable Riemannian space are the Lie point symmetries for geodesic equations of the n-dimensional subspace. We demonstrate the application of our results with the presentation of applications.

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Tsamparlis, Michael, and Andronikos Paliathanasis. "Lie symmetries of geodesic equations and projective collineations." Nonlinear Dynamics 62, no. 1-2 (April 15, 2010): 203–14. http://dx.doi.org/10.1007/s11071-010-9710-x.

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Afonin, K. F. "The method of using differential amendments to convert spatial rectangulars coordinates into those geodesic ones." Geodesy and Cartography 970, no. 4 (May 20, 2021): 2–7. http://dx.doi.org/10.22389/0016-7126-2021-970-4-2-7.

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The article is devoted to considering the technological aspects of the application of differential amendments for the transition from spatial rectangular coordinates to geodesic. It is shown that the coordinate provision of the territories is impossible to imagine at present without the use of GNSS technologies. But they allow to get only spatial rectangular coordinates of defined points. It is noted how the vast majority of users need other coordinates-flat rectangular coordinates in the projection of Gauss–Kruger, which can be calculated only by geodesic latitudes and longitudes. The national and foreign literature describes more than a dozen ways to calculate geodesic coordinates by spatial rectangular coordinates. These methods can be divided into three groups. The first group uses iterative methods and algorithms, the second group uses non-iteter methods. The third group is characterized by the calculation and introduction of amendments to the approximate value of geodesic latitude. It is possible to reduce the number of technological operations that allow to solve the problem without losing the accuracy of calculated geodesic coordinates. There are numerical examples showing the feasibity of the proposed technology at maximum ground points.

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Pişcoran, Laurian-Ioan, and Vishnu Narayan Mishra. "Projective flatness of a new class of ( α , β ) (\alpha,\beta) -metrics." Georgian Mathematical Journal 26, no. 1 (March 1, 2019): 133–39. http://dx.doi.org/10.1515/gmj-2017-0034.

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Abstract In this paper we investigate a new {(\alpha,\beta)} -metric {F=\beta+\frac{a\alpha^{2}+\beta^{2}}{\alpha}} , where {\alpha=\sqrt{{a_{ij}y^{i}y^{j}}}} is a Riemannian metric; {\beta=b_{i}y^{i}} is a 1-form and {a\in(\frac{1}{4},+\infty)} is a real scalar. Also, we investigate the relationship between the geodesic coefficients of the metric F and the corresponding geodesic coefficients of the metric α.

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Andruchow, Esteban. "A note on geodesics of projections in the Calkin algebra." Archiv der Mathematik 115, no. 5 (August 11, 2020): 545–53. http://dx.doi.org/10.1007/s00013-020-01509-5.

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Ximin, Liu. "Totally real submanifolds in a complex projective space." International Journal of Mathematics and Mathematical Sciences 22, no. 1 (1999): 205–8. http://dx.doi.org/10.1155/s0161171299222053.

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In this paper, we establish the following result: LetMbe ann-dimensional complete totally real minimal submanifold immersed inCPnwith Ricci curvature bounded from below. Then eitherMis totally geodesic orinf r≤(3n+1)(n−2)/3, whereris the scalar curvature ofM.

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Watanabe, Yohsuke. "Distances and Intersections of Curves." International Mathematics Research Notices 2020, no. 23 (November 14, 2018): 9674–93. http://dx.doi.org/10.1093/imrn/rny265.

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Abstract We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we study intersection numbers of curves contained in geodesics in the curve graph. Furthermore, we generalize a well-known result on intersection number growth of curves under iteration of Dehn twists and multitwists for all kinds of pure mapping classes.

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