In geometry, a geodesic is a curve representing in some sense the shortest 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".
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.
A locally shortest path between two given points in a curved space, assumed to be a Riemannian manifold, can be defined by using the equation for the length of a curve (a function f from an open interval of R to the space), and then minimizing this length between the points using the calculus of variations. This has some minor technical problems because there is an infinite-dimensional space of different ways to parameterize the shortest path. It is simpler to restrict the set of curves to those that are parameterized "with constant speed" 1, meaning that the distance from f(s) to f(t) along the curve equals |s-t|. Equivalently, a different quantity may be used, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimization). Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its width, and in so doing will minimize its energy. The resulting shape of the band is a geodesic.
It is possible that several different curves between two points minimize the distance, as is the case for two diametrically opposite points on a sphere. In such a case, any of these curves is a geodesic.
A contiguous segment of a geodesic is again a geodesic.
In general, 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 parameterized with "constant speed". 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\tot2
Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry. In general relativity, geodesics in spacetime 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 sub-Riemannian 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 manifolds. The article Levi-Civita connection discusses the more general case of a pseudo-Riemannian manifold and geodesic (general relativity) discusses the special case of general relativity in greater detail.
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 arc of the great circle passing through A and B. If A and B are antipodal points, then there are infinitely many shortest paths between them. Geodesics on an ellipsoid behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).
See also: Toponogov's theorem. A geodesic triangle is formed by the geodesics joining each pair out of three points on a given surface. On the sphere, the geodesics are great circle arcs, forming a spherical triangle.
In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve from an interval I of the reals to the metric space M is a geodesic if there is a constant such that for any there is a neighborhood J of t in I such that for any we have
d(\gamma(t1),\gamma(t2))=v\left|t1-t2\right|.
This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with natural parameterization, i.e. in the above identity v = 1 and
d(\gamma(t1),\gamma(t2))=\left|t1-t2\right|.
If the last equality is satisfied for all, 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. The metric Hopf-Rinow theorem provides situations where a length space is automatically a geodesic space.
Common examples of geodesic metric spaces that are often not manifolds include metric graphs, (locally compact) metric polyhedral complexes, infinite-dimensional pre-Hilbert spaces, and real trees.
In a Riemannian manifold M with metric tensor g, the length L of a continuously differentiable curve γ : [''a'',''b''] → M is defined by
| b | |
| L(\gamma)=\int | |
| a |
\sqrt{g\gamma(t)(
|
}dt.
| E(\gamma)= | 1 |
| 2 |
| b | |
| \int | |
| a |
g\gamma(t)(
|
C1
W1,2
L(\gamma)2\le2(b-a)E(\gamma)
g(\gamma',\gamma')
E(\gamma)
L(\gamma)
\gamma
L(\gamma)
The Euler–Lagrange equations of motion for the functional E are then given in local coordinates by
| d2xλ | |
| dt2 |
+
| λ | |
| \Gamma | |
| \mu\nu |
| dx\mu | |
| dt |
| dx\nu | |
| dt |
=0,
| λ | |
| \Gamma | |
| \mu\nu |
Techniques of the classical calculus of variations can be applied to examine the energy functional E. The first variation of energy is defined in local coordinates by
\deltaE(\gamma)(\varphi)=\left.
| \partial | |
| \partialt |
\right|t=0E(\gamma+t\varphi).
The critical points of the first variation are precisely the geodesics. The second variation is defined by
\delta2E(\gamma)(\varphi,\psi)=\left.
| \partial2 | |
| \partials\partialt |
\right|s=t=0E(\gamma+t\varphi+s\psi).
In an appropriate sense, zeros of the second variation along a geodesic γ arise along Jacobi fields. Jacobi fields are thus regarded as variations through geodesics.
By applying variational techniques from classical mechanics, one can also regard geodesics as Hamiltonian flows. They are solutions of the associated Hamilton equations, with (pseudo-)Riemannian metric taken as Hamiltonian.
See also: Geodesics in general relativity. 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, soat each point along the curve, where
| \gamma |
t
| \gamma |
| \gamma |
Using local coordinates on M, we can write the geodesic equation (using the summation convention) as
| d2\gammaλ | |
| dt2 |
+
| λ | |
| \Gamma | |
| \mu\nu |
| d\gamma\mu | |
| dt |
| d\gamma\nu | |
| dt |
=0 ,
\gamma\mu=x\mu\circ\gamma(t)
| λ | |
| \Gamma | |
| \mu\nu |
| \nabla | |||
|
| \gamma= |
0
The local existence and uniqueness theorem for geodesics states that geodesics on a smooth manifold with an affine connection exist, and are unique. 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
\gamma(0)=p
| \gamma(0) |
=V,
where I is a maximal open interval in R containing 0.
The proof of this theorem follows from the theory of ordinary differential equations, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from the Picard - Lindelöf theorem for the solutions of ODEs with prescribed initial conditions. γ depends smoothly on both p and V.
In general, I may not be all of R as for example for an open disc in R2. Any extends to all of if and only if is geodesically complete.
Geodesic flow is a local R-action on the tangent bundle TM of a manifold M defined in the following way
| |||
| G | |||
| V(t) |
where t ∈ R, V ∈ TM and
\gammaV
| \gamma |
V(0)=V
Gt(V)=\exp(tV)
On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a Hamiltonian flow on the cotangent bundle. The Hamiltonian is then given by the inverse of the (pseudo-)Riemannian metric, evaluated against the canonical one-form. In particular the flow preserves the (pseudo-)Riemannian metric
g
g(Gt(V),Gt(V))=g(V,V).
In particular, when V is a unit vector,
\gammaV
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.
More precisely, an affine connection gives rise to a splitting of the double tangent bundle TTM into horizontal and vertical bundles:
TTM=H ⊕ V.
\pi*Wv=v
More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle TM \ ) it is enough that the connection be equivariant under positive rescalings: it need not be linear. That is, (cf. Ehresmann connection#Vector bundles and covariant derivatives) it is enough that the horizontal distribution satisfy
Hλ=d(Sλ)XHX
Sλ:X\mapstoλX.
Equation is invariant under affine reparameterizations; that is, parameterizations of the form
t\mapstoat+b
An affine connection is determined by its family of affinely parameterized geodesics, up to torsion . The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection. More precisely, if
\nabla,\bar{\nabla}
D(X,Y)=\nablaXY-\bar{\nabla}XY
\nabla
\bar{\nabla}
\nabla
Geodesics without a particular parameterization are described by a projective connection.
Efficient solvers for the minimal geodesic problem on surfaces have been proposed by Mitchell,[1] Kimmel,[2] Crane,[3] and others.
A ribbon "test" is a way of finding a geodesic on a physical surface.[4] 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)\toS
N
\ell
S
f
\ell
\varepsilon
l
| 2) | |
| g | |
| S)=O(\varepsilon |
gN
gS
N
S
While geometric in nature, the idea of a shortest path is so general that it easily finds extensive use in nearly all sciences, and in some other disciplines as well.
Geodesics serve as the basis to calculate: