Geodesic
In
The noun
In a
Geodesics are of particular importance in
Introduction
A locally shortest path between two given points in a curved space, assumed
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 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
This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of
Examples
The most familiar examples are the straight lines in
Triangles
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
Metric geometry
In
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
If the last equality is satisfied for all t1, t2 ∈ 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
Riemannian geometry
In a Riemannian manifold M with metric tensor g, the length L of a continuously differentiable curve γ : [a,b] → M is defined by
The distance d(p, q) between two points p and q of M is defined as the
All minima of E are also minima of L, but L is a bigger set since paths that are minima of L can be arbitrarily re-parameterized (without changing their length), while minima of E cannot. For a piecewise curve (more generally, a curve), the Cauchy–Schwarz inequality gives
with equality if and only if is equal to a constant a.e.; the path should be travelled at constant speed. It happens that minimizers of also minimize , 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 E is a more robust variational problem. Indeed, E is a "convex function" of , 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 are generally not very regular, because arbitrary reparameterizations are allowed.
The Euler–Lagrange equations of motion for the functional E are then given in local coordinates by
where are the Christoffel symbols of the metric. This is the geodesic equation, discussed below.
Calculus of variations
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
The critical points of the first variation are precisely the geodesics. The second variation is defined by
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
Affine geodesics
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, so
-
(1)
at each point along the curve, where is the derivative with respect to . More precisely, in order to define the covariant derivative of it is necessary first to extend to a continuously differentiable vector field in an open set. However, the resulting value of (1) is independent of the choice of extension.
Using
where are the coordinates of the curve γ(t) and are the
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 TpM (the tangent space to M at p) there exists a unique geodesic : I → M such that
- and
- where I is a maximal open intervalin R containing 0.
The proof of this theorem follows from the theory of
In general, I may not be all of R as for example for an open disc in R2. Any γ extends to all of ℝ if and only if M is geodesically complete.
Geodesic flow
Geodesic
where t ∈ R, V ∈ TM and denotes the geodesic with initial data . Thus, is the exponential map of the vector tV. A closed orbit of the geodesic flow corresponds to a closed geodesic on M.
On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a
In particular, when V is a unit vector, remains unit speed throughout, so the geodesic flow is tangent to the unit tangent bundle. Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.
Geodesic spray
The geodesic flow defines a family of curves in the
More precisely, an affine connection gives rise to a splitting of the
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.
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 \ {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
for every X ∈ TM \ {0} and λ > 0. Here d(Sλ) is the pushforward along the scalar homothety A particular case of a non-linear connection arising in this manner is that associated to a Finsler manifold.
Affine and projective geodesics
Equation (1) is invariant under affine reparameterizations; that is, parameterizations of the form
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 (1) are called geodesics with affine parameter.
An affine connection is determined by its family of affinely parameterized geodesics, up to torsion (Spivak 1999, 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 are two connections such that the difference tensor
is skew-symmetric, then and have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as , 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,[3] Kimmel,[4] Crane,[5] and others.
Ribbon test
A ribbon "test" is a way of finding a geodesic on a physical surface.[6] 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 of a
Applications
This section needs expansion. You can help by adding to it. (June 2014) |
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
- Introduction to the mathematics of general relativity – non-technical introduction to the mathematics of general relativity
- Clairaut's relation– Formula in classical differential geometry
- Differentiable curve – Study of curves from a differential point of view
- Differential geometry of surfaces
- Geodesic circle
- Hopf–Rinow theorem – Gives equivalent statements about the geodesic completeness of Riemannian manifolds
- Intrinsic metric – Concept in geometry/topology
- Isotropic line
- Jacobi field
- Morse theory – Analyzes the topology of a manifold by studying differentiable functions on that manifold
- Zoll surface – Surface homeomorphic to a sphere
- The spider and the fly problem – Recreational geodesics problem
Notes
- ^ Lorentzian manifold, the definition is more complicated.
- ^ The path is a local maximum of the interval k rather than a local minimum.
References
- ^ "geodesic". Lexico UK English Dictionary. Oxford University Press. Archived from the original on 2020-03-16.
- ^ "geodesic". Merriam-Webster.com Dictionary.
- doi:10.1137/0216045.
- (PDF) from the original on 2022-10-09.
- S2CID 7078650.
- ^ Michael Stevens (Nov 2, 2017), [1].
- ISBN 978-0-914098-71-3
Further reading
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (July 2014) |
- Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975), Introduction to General Relativity (2nd ed.), New York: ISBN 978-0-07-000423-8. See chapter 2.
- ISBN 978-0-8053-0102-1. See section 2.7.
- Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis, Berlin, New York: ISBN 978-3-540-42627-1. See section 1.4.
- Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. 1 (New ed.), Wiley-Interscience, ISBN 0-471-15733-3.
- ISBN 978-0-08-018176-9. See section 87.
- ISBN 978-0-7167-0344-0
- Ortín, Tomás (2004), Gravity and strings, ISBN 978-0-521-82475-0. Note especially pages 7 and 10.
- Volkov, Yu.A. (2001) [1994], "Geodesic line", Encyclopedia of Mathematics, EMS Press.
- ISBN 978-0-471-92567-5. See chapter 3.
External links
- Geodesics Revisited — Introduction to geodesics including two ways of derivation of the equation of geodesic with applications in geometry (geodesic on a sphere and on a brachistochrone) and optics (light beam in inhomogeneous medium).
- Totally geodesic submanifold at the Manifold Atlas