View source for Geodesic

{{Short description|Straight path on a curved surface or a Riemannian manifold}}
{{About|geodesics in general|geodesics in general relativity|Geodesic (general relativity)|the study of Earth's shape|Geodesy|the application on Earth|Earth geodesic|other uses}}
[[File:Klein quartic with closed geodesics.svg|thumb|[[Klein quartic]] with 28 geodesics <small>(marked by 7 colors and 4 patterns)</small>]]
In [[geometry]], a '''geodesic''' ({{IPAc-en|ˌ|dʒ|iː|.|ə|ˈ|d|ɛ|s|ɪ|k|,_|-|oʊ|-|,_|-|ˈ|d|iː|s|ɪ|k|,_|-|z|ɪ|k}}){{refn|{{Cite dictionary |url=http://www.lexico.com/definition/geodesic |archive-url=https://web.archive.org/web/20200316193343/https://www.lexico.com/definition/geodesic |url-status=dead |archive-date=2020-03-16 |title=geodesic |dictionary=[[Lexico]] UK English Dictionary |publisher=[[Oxford University Press]]}} }}{{refn|{{cite Merriam-Webster|geodesic}}}} is a [[curve]] representing in some sense the locally{{efn|For two points on a sphere that are not antipodes, there are two great circle arcs of different lengths connecting them, both of which are geodesics.}} shortest{{efn|For a [[pseudo-Riemannian manifold]], e.g., a [[Lorentzian manifold]], the definition is more complicated.|name=pseudo}} path ([[arc (geometry)|arc]]) between two points in a [[differential geometry of surfaces|surface]], or more generally in a [[Riemannian manifold]]. The term also has meaning in any  [[differentiable manifold]] with a [[connection (mathematics)|connection]]. It is a generalization of the notion of a "[[Line (geometry)|straight line]]".

The noun ''[[wikt:geodesic|geodesic]]'' and the adjective ''[[wikt:geodetic|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 geodesic|ellipsoidal]] geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's [[Planetary surface|surface]]. For a [[spherical Earth]], it is a [[line segment|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 [[Distance (graph theory)|geodesic]] between two [[vertex (graph theory)|vertices]]/nodes of a [[Graph (discrete mathematics)|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 space|tangent vector]]s remain parallel if they are [[parallel transport|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 fall]]ing [[test particles]].

==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|right|200px|
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|150px|{{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.]]

==Metric geometry==
In [[metric geometry]], a geodesic is a curve which is everywhere [[locally]] a [[distance]] minimizer. More precisely, a [[curve]] {{nowrap|''γ'' : ''I'' → ''M''}} from an interval ''I'' of the reals to the [[metric space]] ''M'' is a '''geodesic''' if there is a [[mathematical constant|constant]] {{nowrap|''v'' ≥ 0}} such that for any {{nowrap|''t'' ∈ ''I''}} there is a neighborhood ''J'' of ''t'' in ''I'' such that for any {{nowrap|''t''<sub>1</sub>, ''t''<sub>2</sub> ∈ ''J''}} we have

:<math>d(\gamma(t_1),\gamma(t_2)) = v \left| t_1 - t_2 \right| .</math>

This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with [[Curve#Lengths of curves|natural parameterization]], i.e. in the above identity ''v''&nbsp;=&nbsp;1 and

:<math>d(\gamma(t_1),\gamma(t_2)) = \left| t_1 - t_2 \right| .</math>

If the last equality is satisfied for all {{nowrap|''t''<sub>1</sub>, ''t''<sub>2</sub> ∈ ''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 path]]s, although this minimizing sequence need not converge to a geodesic. The [[Hopf–Rinow theorem#Variations and generalizations|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 graph|metric graphs]], (locally compact) metric [[Polyhedral complex|polyhedral complexes]], infinite-dimensional [[Pre-Hilbert space|pre-Hilbert spaces]], and [[Real tree|real trees]].

==Riemannian geometry==
In a [[Riemannian manifold]] <math>M</math> with [[metric tensor]] <math>g</math>, the length <math>L</math> of a continuously differentiable curve <math>\gamma : [a,b] \to M</math> is defined by
:<math>L(\gamma)=\int_a^b \sqrt{  g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) }\,dt.</math>
The distance <math>d(p,q)</math> between two points <math>p</math> and <math>q</math> of <math>M</math> is defined as the [[infimum]] of the length taken over all continuous, piecewise continuously differentiable curves <math>\gamma : [a,b]\to M</math> such that <math>\gamma(a)=p</math> and <math>\gamma(b)=q</math>. In Riemannian geometry, all geodesics are locally distance-minimizing paths, but the converse is not true. In fact, only paths that are both locally distance minimizing and parameterized proportionately to arc-length are geodesics. 

Another equivalent way of defining geodesics on a Riemannian manifold, is to define them as the minima of the following [[action (physics)|action]] or [[energy functional]]
:<math>E(\gamma)=\frac{1}{2}\int_a^b g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))\,dt.</math>
All minima of <math>E</math> are also minima of <math>L</math>, but <math>L</math> is a bigger set since paths that are minima of <math>L</math> can be arbitrarily re-parameterized (without changing their length), while minima of <math>E</math> cannot.
For a piecewise <math>C^1</math> curve (more generally, a <math>W^{1,2}</math> curve), the [[Cauchy–Schwarz inequality]] gives
:<math>L(\gamma)^2 \le 2(b-a)E(\gamma)</math>
with equality if and only if <math>g(\gamma',\gamma')</math> is equal to a constant a.e.; the path should be travelled at constant speed.  It happens that minimizers of <math>E(\gamma)</math> also minimize <math>L(\gamma)</math>, because they turn out to be affinely parameterized, and the inequality is an equality.  The usefulness of this approach is that the problem of seeking minimizers of <math>E</math> is a more robust variational problem.  Indeed, <math>E(\gamma)</math> is a "convex function" of <math>\gamma</math>, so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers.  In contrast, "minimizers" of the functional <math>L(\gamma)</math> are generally not very regular, because arbitrary reparameterizations are allowed.

The [[Euler–Lagrange equation]]s of motion for the functional <math>E</math> are then given in local coordinates by
:<math>\frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0,</math>
where <math>\Gamma^\lambda_{\mu\nu}</math> are the [[Christoffel symbols]] of the metric.  This is the '''geodesic equation''', discussed [[#Affine geodesics|below]].

===Calculus of variations===

Techniques of the classical [[calculus of variations]] can be applied to examine the energy functional <math>E</math>.  The [[first variation]] of energy is defined in local coordinates by

:<math>\delta E(\gamma)(\varphi) = \left.\frac{\partial}{\partial t}\right|_{t=0} E(\gamma + t\varphi).</math>

The [[critical point (mathematics)|critical point]]s of the first variation are precisely the geodesics.  The [[second variation]] is defined by

:<math>\delta^2 E(\gamma)(\varphi,\psi) = \left.\frac{\partial^2}{\partial s \, \partial t} \right|_{s=t=0} E(\gamma + t\varphi + s\psi).</math>

In an appropriate sense, zeros of the second variation along a geodesic <math>\gamma</math> arise along [[Jacobi field]]s.  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 equation]]s, with [[Pseudo Riemannian metric|(pseudo-)Riemannian metric]] taken as [[Hamiltonian mechanics|Hamiltonian]].

==Affine geodesics==
{{See also|Geodesics in general relativity}}
A '''geodesic''' on a [[Differentiable manifold|smooth manifold]] <math>M</math> with an [[affine connection]] <math>\nabla</math> is defined as a [[curve]] <math>\gamma(t)</math> such that [[parallel transport]] along the curve preserves the tangent vector to the curve, so
{{NumBlk|:|<math> \nabla_{\dot\gamma} \dot\gamma= 0</math>|{{EquationRef|1}}}}
at each point along the curve,  where <math>\dot\gamma</math> is the derivative with respect to <math>t</math>.  More precisely, in order to define the covariant derivative of <math>\dot\gamma</math> it is necessary first to extend <math>\dot\gamma</math> to a continuously differentiable [[vector field]] in an [[open set]].  However, the resulting value of ({{EquationNote|1}}) is independent of the choice of extension.

Using [[local coordinates]] on <math>M</math>, we can write the '''geodesic equation''' (using the [[summation convention]]) as
:<math>\frac{d^2\gamma^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{d\gamma^\mu }{dt}\frac{d\gamma^\nu }{dt} = 0\ ,</math>
where <math>\gamma^\mu = x^\mu \circ \gamma (t)</math> are the coordinates of the curve <math>\gamma(t)</math> and <math>\Gamma^{\lambda }_{\mu \nu }</math> are the [[Christoffel symbol]]s of the connection <math>\nabla</math>.  This is an [[ordinary differential equation]] for the coordinates.  It has a unique solution, given an initial position and an initial velocity.  Therefore, from the point of view of [[classical mechanics]], geodesics can be thought of as trajectories of [[free particle]]s in a manifold. Indeed, the equation <math> \nabla_{\dot\gamma} \dot\gamma= 0</math> means that the [[Acceleration (differential geometry)|acceleration vector]] of the curve has no components in the direction of the surface (and therefore it is perpendicular to the tangent plane of the surface at each point of the curve). So, the motion is completely determined by the bending of the surface. This is also the idea of general relativity where particles move on geodesics and the bending is caused by gravity.

===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.  More precisely:

:For any point ''p'' in ''M'' and for any vector ''V'' in ''T<sub>p</sub>M'' (the [[tangent space]] to ''M'' at ''p'') there exists a unique geodesic <math>\gamma \,</math> : ''I'' &rarr; ''M'' such that
::<math>\gamma(0) = p \,</math> and
::<math>\dot\gamma(0) = V,</math>
:where ''I'' is a maximal [[open interval]] in '''R''' containing 0.

The proof of this theorem follows from the theory of  [[ordinary differential equation]]s, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from the [[Picard&ndash;Lindelöf theorem]] for the solutions of ODEs with prescribed initial conditions. γ depends [[smooth function|smoothly]] on both ''p'' and&nbsp;''V''.

In general, ''I'' may not be all of '''R''' as for example for an open disc in '''R'''<sup>2</sup>. Any {{mvar|γ}} extends to all of {{mvar|ℝ}} if and only if {{mvar|M}} is [[geodesic manifold|geodesically complete]].

===Geodesic flow{{anchor|Flow}}===
'''Geodesic [[Flow (mathematics)|flow]]''' is a local '''R'''-[[Group action (mathematics)|action]] on the [[tangent bundle]] ''TM'' of a manifold ''M'' defined in the following way

:<math>G^t(V)=\dot\gamma_V(t)</math>

where ''t''&nbsp;∈&nbsp;'''R''', ''V''&nbsp;∈&nbsp;''TM'' and <math>\gamma_V</math> denotes the geodesic with initial data <math>\dot\gamma_V(0)=V</math>. Thus,  ''<math>G^t(V)=\exp(tV)</math> is the [[exponential map (Riemannian geometry)|exponential map]] of the vector ''tV''.  A closed orbit of the geodesic flow corresponds to a [[closed geodesic]] on&nbsp;''M''.

On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a [[Hamiltonian flow]] on the cotangent bundle.  The [[Hamiltonian mechanics|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 <math>g</math>, i.e.

: <math>g(G^t(V),G^t(V))=g(V,V). \, </math>

In particular, when ''V'' is a unit vector, <math>\gamma_V</math> remains unit speed throughout, so the geodesic flow is tangent to the [[unit tangent bundle]].  [[Liouville's theorem (Hamiltonian)|Liouville's theorem]] implies invariance of a kinematic measure on the unit tangent bundle.

===Geodesic spray===
{{further|Spray (mathematics)#Geodesic}}
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 (mathematics)|spray]]'''.

More precisely, an affine connection gives rise to a splitting of the [[double tangent bundle]] TT''M'' into [[horizontal bundle|horizontal]] and [[vertical bundle]]s:
:<math>TTM = H\oplus V.</math>
The geodesic spray is the unique horizontal vector field ''W'' satisfying
:<math>\pi_* W_v = v\,</math>
at each point ''v''&nbsp;∈&nbsp;T''M''; here {{pi}}<sub>∗</sub>&nbsp;:&nbsp;TT''M''&nbsp;→&nbsp;T''M'' denotes the [[pushforward (differential)]] along the projection {{pi}}&nbsp;:&nbsp;T''M''&nbsp;→&nbsp;''M'' associated to the tangent bundle.

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 T''M''&nbsp;\&nbsp;{0}) 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
:<math>H_{\lambda X} = d(S_\lambda)_X H_X\,</math>
for every ''X''&nbsp;∈&nbsp;T''M''&nbsp;\&nbsp;{0} and λ&nbsp;>&nbsp;0.  Here ''d''(''S''<sub>λ</sub>) is the [[pushforward (differential)|pushforward]] along the scalar homothety <math>S_\lambda: X\mapsto \lambda X.</math>  A particular case of a non-linear connection arising in this manner is that associated to a [[Finsler manifold]].

===Affine and projective geodesics===
Equation ({{EquationNote|1}}) is invariant under affine reparameterizations; that is, parameterizations of the form
:<math>t\mapsto at+b</math>
where ''a'' and ''b'' are constant real numbers. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves. Accordingly, solutions of ({{EquationNote|1}}) are called geodesics with '''affine parameter'''.

An affine connection is ''determined by'' its family of affinely parameterized geodesics, up to [[torsion tensor|torsion]] {{harv|Spivak|1999|loc=Chapter 6, Addendum I}}.  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 <math>\nabla, \bar{\nabla}</math> are two connections such that the difference tensor
:<math>D(X,Y) = \nabla_XY-\bar{\nabla}_XY</math>
is [[skew-symmetric matrix|skew-symmetric]], then <math>\nabla</math> and <math>\bar{\nabla}</math> have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as  <math>\nabla</math>, but with vanishing torsion.

Geodesics without a particular parameterization are described by a [[projective connection]].

==Computational methods==
Efficient solvers for the minimal geodesic problem on surfaces have been proposed by Mitchell,<ref>{{cite journal |first1=J. |last1=Mitchell |first2=D. |last2=Mount|first3=C.|last3=Papadimitriou|url=https://epubs.siam.org/doi/10.1137/0216045 |title=The Discrete Geodesic Problem |journal=[[SIAM Journal on Computing]] |volume=16 |issue=4 |pages=647–668 |year=1987 |doi=10.1137/0216045 }}</ref> Kimmel,<ref>{{cite journal |first1=R. |last1=Kimmel |first2=J. A. |last2=Sethian|url=https://www.pnas.org/content/pnas/95/15/8431.full.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.pnas.org/content/pnas/95/15/8431.full.pdf |archive-date=2022-10-09 |url-status=live|title=Computing Geodesic Paths on Manifolds |journal=[[Proceedings of the National Academy of Sciences]] |volume=95 |issue=15 |pages=8431–8435 |year=1998 |doi=10.1073/pnas.95.15.8431 |pmid=9671694 |pmc=21092 |bibcode=1998PNAS...95.8431K |doi-access=free }}</ref> Crane,<ref>{{cite journal |first1=K. |last1=Crane |first2=C. |last2=Weischedel|first3=M. |last3=Wardetzky|url=https://dl.acm.org/doi/10.1145/3131280 |title=The Heat Method for Distance Computation |journal=[[Communications of the ACM]] |volume=60 |issue=11 |pages=90–99 |year=2017 |doi=10.1145/3131280 |s2cid=7078650 }}</ref> and others.

== Ribbon test ==
{{Multiple image
| image1 = 
| caption1 = The Ribbon Test 
| image2 = 
| caption2 = The curved line drawn using the ribbon test is a straight line on a flat surface. This is because a cone can be made into a 2-d circular sector.
}}
A ribbon "test" is a way of finding a geodesic on a physical surface.<ref>[[Michael Stevens (educator)|Michael Stevens]] (Nov 2, 2017), ''[https://www.youtube.com/watch?v=Xc4xYacTu-E]''.</ref> 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 <math>f: N(\ell) \to S</math> of a [[neighborhood_(mathematics)|neighborhood]] <math>N</math> of a line <math>\ell</math> in a plane into a surface <math>S</math> so that the mapping <math>f</math> "doesn't change the distances around <math>\ell</math> by much"; that is, at the distance <math>\varepsilon</math> from <math>l</math> we have <math>g_N-f^*(g_S)=O(\varepsilon^2)</math> where <math>g_N</math> and <math>g_S</math> are [[metric_tensor|metrics]] on <math>N</math> and <math>S</math>.

==Examples of applications==
{{expand section|date=June 2014}}
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.

=== Topology and geometric group theory ===

* In a surface with negative [[Euler characteristic]], any (free) homotopy class determines a unique (closed) geodesic for a [[Hyperbolic surface|hyperbolic]] metric. These geodesics contribute significantly to the geometric understanding of the action of [[Mapping class group of a surface|mapping classes]].
* [[Geodesic metric space|Geodesic metric spaces]] and [[Length space|length spaces]] behave particularly well with isometric [[Group action|group actions]] ([[Švarc–Milnor lemma|Švarc-Milnor lemma]], [[Hopf–Rinow theorem#Variations and generalizations|Hopf-Rinow theorem]], [[Quasi-isometry#Quasigeodesics and the Morse lemma|Morse lemma]]...). They are often an adequate framework for generalizing results from Riemannian geometry to constructions that reflect the geometry of a group. For instance, [[Hyperbolic metric space|Gromov-hyperbolicity]] can be understood in terms of geodesic triangle thinness, and [[CAT(k) space|CAT(0)]] can be stated in terms of angles between geodesics.

=== Probability, statistics and machine learning ===

* [[Optimal transport]] can be understood as the problem of finding geodesic paths in spaces of measures.
* In [[information geometry]], [[Divergence (statistics)|divergences]] such as the [[Kullback–Leibler divergence|Kullback-Leibler divergence]] play a role analogous to that of a Riemannian metric, allowing analogies for [[Metric connection|connections]] and geodesics.

=== Physics ===

* In [[classical mechanics]], [[Trajectory|trajectories]] minimize an energy according to the [[Hamilton–Jacobi equation|Hamilton-Jacobi equation]], which can re regarded as a similar idea to geodesics. In some special cases, [[Geodesics as Hamiltonian flows|the two notions actually coincide]].
* [[Theory of relativity|Relativity theory]] models [[spacetime]] as a [[Lorentzian manifold]], where light follows Lorentzian geodesics.

=== Biology ===

* The study of how the [[nervous system]] optimizes muscular movement may be approached by endowing a [[Configuration space (physics)|configuration space]] of the body with a [[Riemannian metric]] that measures the effort, so that the problem can be stated in terms of geodesy.<ref>{{Cite journal |last1=Neilson |first1=Peter D. |last2=Neilson |first2=Megan D. |last3=Bye |first3=Robin T. |date=2015-12-01 |title=A Riemannian geometry theory of human movement: The geodesic synergy hypothesis |url=https://www.sciencedirect.com/science/article/abs/pii/S0167945715300208 |journal=Human Movement Science |volume=44 |pages=42–72 |doi=10.1016/j.humov.2015.08.010 |pmid=26302481 |issn=0167-9457}}</ref>
* [[Geodesic distance]] is often used to measure the length of paths for signal propagation in neurons.<ref>{{Cite journal |last1=Beshkov |first1=Kosio |last2=Tiesinga |first2=Paul |date=2022-02-01 |title=Geodesic-based distance reveals nonlinear topological features in neural activity from mouse visual cortex |url=https://link.springer.com/article/10.1007/s00422-021-00906-5 |journal=Biological Cybernetics |language=en |volume=116 |issue=1 |pages=53–68 |doi=10.1007/s00422-021-00906-5 |pmid=34816322 |issn=1432-0770}}</ref>
* The structures of geodesics in large molecules plays a role in the study of [[protein folds]].<ref>{{Cite journal |last1=Zanotti |first1=Giuseppe |last2=Guerra |first2=Concettina |date=2003-01-16 |title=Is tensegrity a unifying concept of protein folds? |url=https://www.sciencedirect.com/science/article/pii/S001457930203853X |journal=FEBS Letters |volume=534 |issue=1 |pages=7–10 |doi=10.1016/S0014-5793(02)03853-X |pmid=12527354 |bibcode=2003FEBSL.534....7Z |issn=0014-5793}}</ref>

=== Engineering ===
Geodesics serve as the basis to calculate:
* geodesic airframes; see [[geodesic airframe]] or [[geodetic 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 [[motion planning]] (e.g., when painting car parts); see [[Shortest path problem]]
* geodesic shortest path (GSP) correction over [[Poisson surface reconstruction]] (e.g. in [[digital dentistry]]); without GSP reconstruction often results in self-intersections within the surface

==See also==
{{div col}}
* {{annotated link|Introduction to the mathematics of general relativity}}
* {{annotated link|Clairaut's relation}}
* {{annotated link|Differentiable curve}}
* [[Differential geometry of surfaces]]
* [[Geodesic circle]]
* {{annotated link|Hopf&ndash;Rinow theorem}}
* {{annotated link|Intrinsic metric}}
* {{annotated link|Isotropic line}}
* {{annotated link|Jacobi field}}
* {{annotated link|Morse theory}}
* {{annotated link|Zoll surface}}
* {{annotated link|The spider and the fly problem}}
{{div col end}}

==Notes==
{{Notelist}}

==References==
{{Reflist}}
*{{Citation | last1=Spivak | first1=Michael | author1-link=Michael Spivak | title=A Comprehensive introduction to differential geometry (Volume 2) | publisher=Publish or Perish | location=Houston, TX | isbn=978-0-914098-71-3 | year=1999}}
{{Commons category|Geodesic (mathematics)}}

==Further reading==
{{more footnotes|date=July 2014}}
*{{Citation | last1=Adler | first1=Ronald | last2=Bazin | first2=Maurice | last3=Schiffer | first3=Menahem | title=Introduction to General Relativity | publisher=[[McGraw-Hill]] | location=New York | edition=2nd | isbn=978-0-07-000423-8 | year=1975}}. ''See chapter 2''.
*{{Citation | last1=Abraham | first1=Ralph H. | author1-link=Ralph Abraham (mathematician) | last2=Marsden | first2=Jerrold E. | author2-link=Jerrold E. Marsden | title=Foundations of mechanics | publisher=Benjamin-Cummings | location=London | isbn=978-0-8053-0102-1 | year=1978}}. ''See section 2.7''.
*{{Citation | last1=Jost | first1=Jürgen | title=Riemannian Geometry and Geometric Analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-42627-1 | year=2002}}. ''See section 1.4''.
*{{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = Foundations of Differential Geometry|volume=1| publisher=Wiley-Interscience | year=1996|edition=New|isbn=0-471-15733-3}}.
*{{Citation | last1=Landau | first1=L. D. | author1-link=Lev Landau | last2=Lifshitz | first2=E. M. | author2-link=Evgeny Lifshitz | title=Classical Theory of Fields | publisher=Pergamon | location=Oxford | isbn=978-0-08-018176-9 | year=1975}}. ''See section 87''.
*{{Citation | last1=Misner | first1=Charles W. | author1-link=Charles W. Misner | last2=Thorne | first2=Kip | author2-link=Kip Thorne | last3=Wheeler | first3=John Archibald | author3-link=John Archibald Wheeler | title=Gravitation | publisher=W. H. Freeman | isbn=978-0-7167-0344-0 | year=1973| title-link=Gravitation (book) }}
*{{Citation | last1=Ortín | first1=Tomás | title=Gravity and strings | publisher=[[Cambridge University Press]] | isbn=978-0-521-82475-0 | year=2004}}.  Note especially pages 7 and 10.
*{{springer|first=Yu.A.|last=Volkov|title=Geodesic line|id=G/g044120}}.
*{{Citation | last1=Weinberg | first1=Steven | author1-link=Steven Weinberg | title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-471-92567-5 | year=1972 | url-access=registration | url=https://archive.org/details/gravitationcosmo00stev_0 }}. ''See chapter 3''.

== External links ==
{{wikiquote}}
* [http://www.cmsim.eu/papers_pdf/january_2012_papers/25_CMSIM_2012_Pokorny_1_281-298.pdf Geodesics Revisited] &mdash; 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).
* [http://www.map.mpim-bonn.mpg.de/Totally_geodesic_submanifold Totally geodesic submanifold] at the Manifold Atlas

{{Riemannian geometry}}
{{Manifolds}}
{{Tensors}}
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[[Category:Differential geometry]]
[[Category:Geodesic (mathematics)| ]]