Editing Geodesic (section)

==Introduction==
A locally shortest path between two given points in a curved space, assumed{{efn|name=pseudo}} to be a [[Riemannian manifold]], can be defined by using the [[equation]] for the [[Arc length|length]] of a [[curve]] (a function ''f'' from an [[open interval]] of '''[[Real number line|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''&minus;''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).{{citation needed|date=May 2018}} 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 <math>t \to t^2</math> 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.

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 particle]]s 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{{efn|The path is a local maximum of the interval ''k'' rather than a local minimum.}} 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 manifold]]s. 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.

===Examples===
[[File:Transpolar geodesic on a triaxial ellipsoid case A.svg|thumb|
A [[geodesics on a triaxial ellipsoid|geodesic on a triaxial ellipsoid]].]]
[[File:Insect on a torus tracing out a non-trivial geodesic.gif|thumb|right|If an insect is placed on a surface and continually walks "forward", by definition it will trace out a geodesic.]]
The most familiar examples are the straight lines in [[Euclidean geometry]]. On a  [[sphere]], the images of geodesics are the  [[great circle]]s. The shortest path from point ''A'' to point ''B'' on a sphere is given by the shorter [[arc (geometry)|arc]] of the great circle passing through ''A'' and ''B''. If ''A'' and ''B'' are [[antipodal point]]s, 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).

===Triangles{{anchor|Triangle}}===
{{see also|Gauss–Bonnet theorem#For triangles|Toponogov's theorem}}
[[File:Spherical triangle.svg|thumb|left|{{Anchor|Triangle}}A geodesic triangle on the sphere.]]
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]].
[[Image:End of universe.jpg|thumb|left|Geodesic triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.]]