Geodesic

In geometry, a geodesic (/ˌdʒiː.əˈdɛsɪk, -oʊ-, -ˈdiːsɪk, -zɪk/)^[1]^[2] is a curve representing in some sense the shortest^[a] path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line".

This article is about geodesics in general. For geodesics in general relativity, see Geodesic (general relativity). For the study of Earth's shape, see Geodesy. For the application on Earth, see Earth geodesic. For other uses, see Geodesic (disambiguation).

The noun geodesic and the adjective geodetic come from geodesy, the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.

In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.

Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles.

Computational methods[edit]

Efficient solvers for the minimal geodesic problem on surfaces have been proposed by Mitchell,^[3] Kimmel,^[4] Crane,^[5] and others.

$_$_$DEEZ_NUTS#0__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#0__subtitleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#7__descriptionDEEZ_NUTS$_$_$

Ribbon test[edit]

A ribbon "test" is a way of finding a geodesic on a physical surface.^[6] The idea is to fit a bit of paper around a straight line (a ribbon) onto a curved surface as closely as possible without stretching or squishing the ribbon (without changing its internal geometry).

For example, when a ribbon is wound as a ring around a cone, the ribbon would not lie on the cone's surface but stick out, so that circle is not a geodesic on the cone. If the ribbon is adjusted so that all its parts touch the cone's surface, it would give an approximation to a geodesic.

Mathematically the ribbon test can be formulated as finding a mapping $f:N(\ell )\to S$ of a neighborhood $N$ of a line $\ell$ in a plane into a surface $S$ so that the mapping $f$ "doesn't change the distances around $\ell$ by much"; that is, at the distance $\varepsilon$ from $l$ we have $g_{N}-f^{*}(g_{S})=O(\varepsilon ^{2})$ where $g_{N}$ and $g_{S}$ are metrics on $N$ and $S$ .

$_$_$DEEZ_NUTS#1__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__descriptionDEEZ_NUTS$_$_$

Geodesics serve as the basis to calculate:

$_$_$DEEZ_NUTS#5__heading--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__description--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__heading--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__description--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__heading--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__description--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__heading--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__description--3DEEZ_NUTS$_$_$

geodesic airframes; see or geodetic airframe

geodesic airframe

geodesic structures – for example

geodesic domes

horizontal distances on or near Earth; see

Earth geodesics

mapping images on surfaces, for rendering; see

UV mapping

robot (e.g., when painting car parts); see Shortest path problem

motion planning

geodesic shortest path (GSP) correction over (e.g. in digital dentistry); without GSP reconstruction often results in self-intersections within the surface

Poisson surface reconstruction

$_$_$DEEZ_NUTS#5__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__descriptionDEEZ_NUTS$_$_$

(1999), A Comprehensive introduction to differential geometry (Volume 2), Houston, TX: Publish or Perish, ISBN 978-0-914098-71-3

Spivak, Michael

Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975), Introduction to General Relativity (2nd ed.), New York: , ISBN 978-0-07-000423-8. See chapter 2.

McGraw-Hill

; Marsden, Jerrold E. (1978), Foundations of mechanics, London: Benjamin-Cummings, ISBN 978-0-8053-0102-1. See section 2.7.

Abraham, Ralph H.

Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis, Berlin, New York: , ISBN 978-3-540-42627-1. See section 1.4.

Springer-Verlag

Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. 1 (New ed.), Wiley-Interscience, 0-471-15733-3.

ISBN

; Lifshitz, E. M. (1975), Classical Theory of Fields, Oxford: Pergamon, ISBN 978-0-08-018176-9. See section 87.

Landau, L. D.

; Thorne, Kip; Wheeler, John Archibald (1973), Gravitation, W. H. Freeman, ISBN 978-0-7167-0344-0

Misner, Charles W.

Ortín, Tomás (2004), Gravity and strings, , ISBN 978-0-521-82475-0. Note especially pages 7 and 10.

Cambridge University Press

Volkov, Yu.A. (2001) [1994], , Encyclopedia of Mathematics, EMS Press.

"Geodesic line"

(1972), Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, New York: John Wiley & Sons, ISBN 978-0-471-92567-5. See chapter 3.

Weinberg, Steven

— Introduction to geodesics including two ways of derivation of the equation of geodesic with applications in geometry (geodesic on a sphere and on a torus), mechanics (brachistochrone) and optics (light beam in inhomogeneous medium).

Geodesics Revisited

at the Manifold Atlas

Totally geodesic submanifold

$_$_$DEEZ_NUTS#9__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--8DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--8DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--8DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__company_or_position--0DEEZ_NUTS$_$_$