Dictionary Definition
geodesic adj
1 of or resembling a geodesic dome
2 of or relating to or determined by geodesy
[syn: geodetic,
geodesical] n :
(mathematics) the shortest line between two points on a
mathematically defined surface (as a straight line on a plane or an
arc of a great circle on a sphere) [syn: geodesic
line]
User Contributed Dictionary
English
Pronunciation
 Rhymes: iːsɪk
Translations
 German: Geodäte
Adjective
 of or relating to geodesy
 of or relating to a geodesic dome
Translations
 German: geodätisch
 Italian: geodetico
Extensive Definition
In mathematics, a geodesic
/ˌdʒiəˈdɛsɪk, ˈdisɪk/[jeeuhdesik,
deesik] is a generalization of the notion of a "straight
line" to "curved
spaces". In presence of a metric,
geodesics are defined to be (locally) the shortest path
between points on the space. In the presence of an affine
connection, geodesics are defined to be curves whose tangent
vectors remain parallel if they are transported
along it.
The term "geodesic" comes from geodesy, the science of
measuring the size and shape of Earth; in the
original sense, a geodesic was the shortest route between two
points on the Earth's surface, namely, a segment of a
great
circle. The term has been generalized to include measurements
in much more general mathematical spaces; for example, in graph
theory, one might consider a geodesic between two
vertices/nodes of a graph.
Geodesics are of particular importance in
general
relativity, as they describe the motion of inertial test
particles.
Introduction
The shortest path between two points in a curved space can be found by writing the equation for the length of a curve (a function f from an open interval of R to the manifold), and then minimizing this length using the calculus of variations. This has some minor technical problems, because there is an infinite dimensional space of different ways to parametrize the shortest path. It is simpler to demand not only that the curve locally minimize length but also that it is parametrized "with constant velocity", meaning that the distance from f(s) to f(t) along the geodesic is proportional tos−t. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a "constant velocity" geodesic. Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy; the resulting shape of the band is a geodesic.In Riemannian geometry geodesics are not the same
as "shortest curves" between two points, though the two concepts
are closely related. The difference is that geodesics are only
locally the shortest distance between points, and are parametrized
with "constant velocity". Going the "long way round" on a great
circle between two points on a sphere is a geodesic but not the
shortest path between the points. The map t→t2 from the
unit interval to itself gives the shortest path between 0 and 1,
but is not a geodesic because the velocity of the corresponding
motion of a point is not constant.
Geodesics are commonly seen in the study of
Riemannian
geometry and more generally metric
geometry. In relativistic physics, geodesics describe the
motion of point
particles under the influence of gravity alone. In particular,
the path taken by a falling rock, an orbiting satellite, or the shape of a
planetary
orbit are all geodesics in curved spacetime. More generally,
the topic of subRiemannian
geometry deals with the paths that objects may take when they
are not free, and their movement is constrained in various
ways.
This article presents the mathematical formalism
involved in defining, finding, and proving the existence of
geodesics, in the case of Riemannian
and pseudoRiemannian
manifolds. The article
geodesic (general relativity) discusses the special case of
general relativity in greater detail.
Examples
The most familiar examples are the straight lines in Euclidean geometry. On a sphere, the images of geodesics are the great circles. The shortest path from point A to point B on a sphere is given by the shorter piece of the great circle passing through A and B. If A and B are antipodal points (like the North pole and the South pole), then there are infinitely many shortest paths between them.Metric geometry
In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve γ: I → M from an interval I of the reals to the metric space M is a geodesic if there is a constant v ≥ 0 such that for any t ∈ I there is a neighborhood J of t in I such that for any t1, t2 ∈ J we have d(\gamma(t_1),\gamma(t_2))=vt_1t_2.\,
This generalizes the notion of geodesic for
Riemannian manifolds. However, in metric geometry the geodesic
considered is often equipped with natural
parametrization, i.e. in the above identity v = 1 and
 d(\gamma(t_1),\gamma(t_2))=t_1t_2.\,
If the last equality is satisfied for all t1, t2
∈I, the geodesic is called a minimizing geodesic or shortest
path.
In general, a metric space may have no geodesics,
except constant curves. At the other extreme, any two points in a
length
metric space are joined by a minimizing sequence of rectifiable
paths, although this minimizing sequence need not converge to a
geodesic.
(Pseudo)Riemannian geometry
A geodesic on a smooth manifold M with an affine connection ∇ is defined as a curve γ(t) such that parallel transport along the curve preserves the tangent vector to the curve, so \nabla_ \dot\gamma= 0
Using local
coordinates on M, we can write the geodesic equation (using the
summation
convention) as
 \frac + \Gamma^_\frac\frac = 0\ ,
Geodesics for a (pseudo)Riemannian
manifold M are defined to be geodesics for its LeviCivita
connection. In a Riemannian manifold a geodesic is the same as
a curve that locally minimizes the length
 l(\gamma)=\int_\gamma \sqrt\,dt\ ,
 S(\gamma)=\frac\int g(\dot\gamma(t),\dot\gamma(t))\,dt,
In a similar manner, one can obtain geodesics as
a solution of the
Hamilton–Jacobi equations, with (pseudo)Riemannian metric
taken as Hamiltonian.
See
Riemannian manifolds in Hamiltonian mechanics for further
details.
Existence and uniqueness
The local existence and uniqueness theorem for geodesics states that geodesics on a smooth manifold with an affine connection exist, and are unique; this is a variant of the Frobenius theorem. More precisely: For any point p in M and for any vector V in TpM (the tangent space to M at p) there exists a unique geodesic \gamma \, : I → M such that

 \gamma(0) = p \, and
 \dot\gamma(0) = V,
 where I is a maximal open interval in R containing 0.
In general, I may not be all of R as for example
for an open disc in R2. The proof of this theorem follows from the
theory of
ordinary differential equations, by noticing that the geodesic
equation is a secondorder ODE. Existence and uniqueness then
follow from the PicardLindelöf
theorem for the solutions of ODEs with prescribed initial
conditions. γ depends smoothly
on both p and V.
Geodesic flow
Geodesic flow is an \mathbb Raction on tangent bundle T(M) of a manifold M defined in the following way G^t(V)=\dot\gamma_V(t)
It defines a Hamiltonian
flow on (co)tangent bundle with the (pseudo)Riemannian metric
as the Hamiltonian. In
particular it preserves the (pseudo)Riemannian metric g, i.e.
 g(G^t(V),G^t(V))=g(V,V).
Geodesic spray
The geodesic flow defines a family of curves in the tangent bundle. The derivatives of these curves define a vector field on the total space of the tangent bundle, known as the geodesic spray.See also
 Basic introduction to the mathematics of curved spacetime
 Complex geodesic
 Differential geometry of curves
 Exponential map
 Geodesic dome
 Geodesic (general relativity)
 Geodesics as Hamiltonian flows
 HopfRinow theorem
 Intrinsic metric
 Jacobi field
 Quasigeodesic
 Solving the geodesic equations
 Barnes Wallis, who applied geodesics to aircraft structural design in the design of the Vickers Wellesley and Vickers Wellington aircraft, and the R100 airship.
References
 Riemannian Geometry and Geometric Analysis . See section 1.4.
 Introduction to General Relativity . See chapter 2.
 Foundations of mechanics . See section 2.7.
 Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity . See chapter 3.
 Classical Theory of Fields . See section 87.
 Gravitation
 Gravity and strings . Note especially pages 7 and 10.
External links
 Caltech Tutorial on Relativity — A nice, simple explanation of geodesics with accompanying animation.
 Geomath
geodesic in Bengali: জিওডেসিক
geodesic in Catalan: Geodèsica
geodesic in German: Geodäte
geodesic in Spanish: Geodésica
geodesic in French: Géodésique
geodesic in Galician: Xeodésica
geodesic in Korean: 측지선
geodesic in Italian: Geodetica
geodesic in Lithuanian: Geodezinė kreivė
geodesic in Dutch: Geodeet (wiskunde)
geodesic in Japanese: 測地線
geodesic in Norwegian: Geodetisk kurve
geodesic in Polish: Linia geodezyjna
geodesic in Portuguese: Geodésica
geodesic in Russian: Геодезическая
geodesic in Simple English: Geodesic
geodesic in Slovenian: Geodetka
geodesic in Serbian: Геодезијска линија
geodesic in Finnish: Geodeesi
geodesic in Ukrainian: Геодезична лінія
geodesic in Chinese: 测地线