Editing Geodesic (section)

==Examples of applications==
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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 be 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