scholarly journals An approach for designing a surface pencil through a given geodesic curve

2021 ◽  
Vol 70 (1) ◽  
pp. 555-568
Author(s):  
Gülnur ŞAFFAK ATALAY ◽  
Fatma GÜLER ◽  
Ergin BAYRAM ◽  
Emin KASAP
Keyword(s):  
Geodesic Curve

Acoustic emission source location in composite structure by Voronoi construction using geodesic curve evolution

2009 ◽  
Vol 126 (5) ◽  
pp. 2324-2330 ◽  
Author(s):  
R. Gangadharan ◽  
G. Prasanna ◽  
M. R. Bhat ◽  
C. R. L. Murthy ◽  
S. Gopalakrishnan
Keyword(s):  
Acoustic Emission ◽  
Composite Structure ◽  
Source Location ◽  
Emission Source ◽  
Curve Evolution ◽  
Geodesic Curve ◽  
Acoustic Emission Source

Metric Spaces and Geodesic Motion

2019 ◽  
pp. 355-384
Author(s):  
David D. Nolte
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Parallel Transport ◽  
Variational Calculus ◽  
Geodesic Equation ◽  
Geodesic Curve ◽  
Geodesic Motion ◽  
Metric Tensor ◽  
Christoffel Symbols ◽  
Derivatives Of

The metric tensor uniquely defines the geometric properties of a metric space, while differential geometry is concerned with the derivatives of vectors and tensors within the metric space. Derivatives of vector quantities include the derivatives of basis vectors, which lead to Christoffel symbols that are directly connected to the metric tensor. This connection between tensor derivatives and the metric tensor provides the tools necessary to define geodesic curves. The geodesic equation is derived from variational calculus and from parallel transport. Geodesic motion is the trajectory of a particle through a metric space defined by an action metric that includes a potential function. In this way, dynamics converts to geometry as a trajectory in a potential converts to a geodesic curve through an appropriately defined metric space.

FINDING THE SHORTEST PATHS ON SURFACES BY FAST GLOBAL APPROXIMATION AND PRECISE LOCAL REFINEMENT

1996 ◽  
Vol 10 (06) ◽  
pp. 643-656 ◽  
Author(s):  
RON KIMMEL ◽  
NAHUM KIRYATI
Keyword(s):  
Shortest Path ◽  
Optimal Path ◽  
Shortest Paths ◽  
Geodesic Curvature ◽  
Geodesic Curve ◽  
Computationally Efficient ◽  
Global Approximation ◽  
Curve Shortening Flow ◽  
Curve Shortening ◽  
Fixed End

Finding the shortest path between points on a surface is a challenging global optimization problem. It is difficult to devise an algorithm that is computationally efficient, locally accurate and guarantees to converge to the globally shortest path. In this paper a two stage coarse-to-fine approach for finding the shortest paths is suggested. In the first stage the algorithm of Ref. 10 that combines a 3D length estimator with graph search is used to rapidly obtain an approximation to the globally shortest path. In the second stage the approximation is refined to become a shorter geodesic curve, i.e., a locally optimal path. This is achieved by using an algorithm that deforms an arbitrary initial curve ending at two given surface points via geodesic curvature shortening flow. The 3D curve shortening flow is transformed into an equivalent 2D one that is implemented using an efficient numerical algorithm for curve evolution with fixed end points, introduced in Ref. 9.

scholarly journals On the Construction of a Surface Family with Common Geodesic in Galilean Space G3

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 360-363 ◽  
Author(s):  
Zühal Küçükarslan Yüzbaşı ◽  
Mehmet Bektaş
Keyword(s):  
Sufficient Conditions ◽  
Parametric Representation ◽  
Necessary And Sufficient Conditions ◽  
Geodesic Curve ◽  
Frenet Frame ◽  
Parametric Surfaces ◽  
Galilean Space ◽  
Necessary And Sufficient

AbstractIn this paper, we investigate the parametric representation for a family of surfaces through a given geodesic curve G3. We provide necessary and sufficient conditions for this curve to be an isogeodesic curve on the parametric surfaces using Frenet frame in Galilean space. Also, for the sake of visualizing of this study, we plot an example for this surfaces family.

scholarly journals Motion of Charged Spinning Particles in a Unified Field

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
M. I. Wanas ◽  
Mona M. Kamal
Keyword(s):  
Field Theory ◽  
Test Particle ◽  
Geodesic Curve ◽  
Electromagnetic Potential ◽  
Unified Field Theory ◽  
Potential Term ◽  
Unified Field ◽  
New Feature ◽  
Two Parameters ◽  
Curve Equation

Using a geometry wider than Riemannian one, the parameterized absolute parallelism (PAP) geometry, we derived a new curve containing two parameters. In the context of the geometrization philosophy, this new curve can be used as a trajectory of charged spinning test particle in any unified field theory constructed in the PAP space. We show that imposing certain conditions on the two parameters, the new curve can be reduced to a geodesic curve giving the motion of a scalar test particle or/and a modified geodesic giving the motion of neutral spinning test particle in a gravitational field. The new method used for derivation, the Bazanski method, shows a new feature in the new curve equation. This feature is that the equation contains the electromagnetic potential term together with the Lorentz term. We show the importance of this feature in physical applications.

Clairaut Riemannian maps whose total manifolds admit a Ricci soliton

2021 ◽  
Author(s):  
Akhilesh Yadav ◽  
Kiran Meena
Keyword(s):  
Vector Field ◽  
Necessary Conditions ◽  
Curvature Vector ◽  
Ricci Soliton ◽  
Necessary Condition ◽  
Geodesic Curve ◽  
Necessary And Sufficient Condition ◽  
Gradient Ricci Soliton ◽  
Riemannian Map ◽  
Necessary And Sufficient

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field [Formula: see text] to be conformal, where [Formula: see text] is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of [Formula: see text] then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation.

scholarly journals Algorithm for filling curves on surfaces

2020 ◽  
Vol 208 (1) ◽  
pp. 49-59
Author(s):  
Monika Kudlinska
Keyword(s):  
Euler Characteristic ◽  
Efficient Method ◽  
Efficient Algorithm ◽  
Orientable Surface ◽  
Hyperbolic Metric ◽  
Geodesic Curve ◽  
Explicit Bound ◽  
Compact Orientable Surface ◽  
Curves On Surfaces

AbstractLet $$\varSigma $$ Σ be a compact, orientable surface of negative Euler characteristic, and let h be a complete hyperbolic metric on $$\varSigma $$ Σ . A geodesic curve $$\gamma $$ γ in $$\varSigma $$ Σ is filling if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word in some generators of $$\pi _1(\varSigma )$$ π 1 ( Σ ) , is filling. In the process, we find an explicit bound for the combinatorial length of a curve given by its Dehn–Thurston coordinate, in terms of the hyperbolic length. This gives us an efficient method for producing a collection which is guaranteed to contain all words corresponding to simple geodesics of bounded hyperbolic length.

scholarly journals Geodesic Computations on Surfaces

2013 ◽  
Author(s):  
Karthik Krishnan
Keyword(s):  
Geodesic Distance ◽  
Front Propagation ◽  
Geodesic Curve ◽  
Fast Marching Method ◽  
Triangle Mesh ◽  
Fast Marching ◽  
Geometry Processing ◽  
Distance Field ◽  
Triangulated Surface ◽  
Termination Criteria

The computation of geodesic distances on a triangle mesh has many applications in geometry processing. The fast marching method provides an approximation of the true geodesic distance field. We provide VTK classes to compute geodesics on triangulated surface meshes. This includes classes for computing the geodesic distance field from a set of seeds and to compute the geodesic curve between source and destination point(s) by back-tracking along the gradient of the distance field. The fast marching toolkit (Peyre et. al.) is internally used. A variety of options are exposed to guide front propagation including the ability to specify propagation weights, constrain to a region, specify exclusion regions, and distance based termination criteria. Interpolators that plug into a contour widget, are provided to enable interactive tracing of paths on meshes.

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