Geodesic

In geometry, a geodesic (/ˌdʒiː.əˈdɛsɪk, -oʊ-, -ˈdiːsɪk, -zɪk/)[1][2] is a curve representing in some sense the locally[a] shortest[b] path (arc) between two points in a surface, or more generally in a Riemannian manifold.

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.

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.

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[b] 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.

[citation needed] 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 difference is that geodesics are only locally the shortest distance between points, and are parameterized with "constant speed".

from the unit interval on the real number line 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.

In general relativity, geodesics in spacetime describe the motion of point particles under the influence of gravity alone.

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.

Geodesics on an ellipsoid behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).

On the sphere, the geodesics are great circle arcs, forming a spherical triangle.

In fact, only paths that are both locally distance minimizing and parameterized proportionately to arc-length are geodesics.

, so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers.

Techniques of the classical calculus of variations can be applied to examine the energy functional

By applying variational techniques from classical mechanics, one can also regard geodesics as Hamiltonian flows.

Therefore, from the point of view of classical mechanics, geodesics can be thought of as trajectories of free particles in a manifold.

This is also the idea of general relativity where particles move on geodesics and the bending is caused by gravity.

Existence and uniqueness then follow from the Picard–Lindelöf theorem for the solutions of ODEs with prescribed initial conditions.

Liouville's theorem implies invariance of a kinematic measure on the unit 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: The geodesic spray is the unique horizontal vector field W satisfying at each point v ∈ TM; here π∗ : TTM → TM denotes the pushforward (differential) along the projection π : TM → M associated to the tangent bundle.

For the resulting vector field to be a spray (on the deleted tangent bundle TM \ {0}) it is enough that the connection be equivariant under positive rescalings: it need not be linear.

Ehresmann connection#Vector bundles and covariant derivatives) it is enough that the horizontal distribution satisfy for every X ∈ TM \ {0} and λ > 0.

Equation (1) is invariant under affine reparameterizations; that is, parameterizations of the form where a and b are constant real numbers.

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

If the ribbon is adjusted so that all its parts touch the cone's surface, it would give an approximation to a geodesic.

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.