, a geodesic () is commonly a curve
representing in some sense the shortest path (arc
) between two points in a surface
, or more generally in a Riemannian manifold
. The term also has meaning in any differentiable manifold
with a connection
. It is a generalization of the notion of a "straight line
" to a more general setting.
The noun "geodesic" and the adjective "geodetic" come from ''geodesy
'', the science of measuring the size and shape of Earth
, while many of the underlying principles can be applied to any ellipsoidal
geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface
. For a spherical Earth
, it is a segment
of a great circle
(see also great-circle distance
). The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory
, one might consider a geodesic
between two vertices
/nodes of a 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 vector
s remain parallel if they are 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
The shortest path between two given points in a curved space, assumed to be a differential manifold
, can be defined by using the equation
for the length
of a curve
(a function ''f'' from an open interval
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''−''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). Intuitively, one can understand this second formulation by noting that an elastic band
stretched between two points will contract its length, 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
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 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.
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
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).
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
left|Geodesic triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.
In metric geometry
, a geodesic is a curve which is everywhere locally
minimizer. More precisely, a curve
from an interval ''I'' of the reals to the metric space
''M'' is a geodesic if there is a constant
such that for any there is a neighborhood ''J'' of ''t'' in ''I'' such that for any we have
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 , 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.
In a Riemannian manifold
''M'' with metric tensor
''g'', the length ''L'' of a continuously differentiable curve γ : 'a'',''b''
nbsp;→ ''M'' is defined by
The distance ''d''(''p'', ''q'') between two points ''p'' and ''q'' of ''M'' is defined as the infimum
of the length taken over all continuous, piecewise continuously differentiable curves γ : 'a'',''b''
nbsp;→ ''M'' such that γ(''a'') = ''p'' and γ(''b'') = ''q''. 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
or energy functional
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
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
, 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 equation
s of motion for the functional ''E'' are then given in local coordinates by
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 point
s 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 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 taken as Hamiltonian
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
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 () is independent of the choice of extension.
Using local coordinates
on ''M'', we can write the geodesic equation (using the summation convention
are the coordinates of the curve γ(''t'') and
are the Christoffel symbol
s of the connection ∇. 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
means that the 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 ''Tp
M'' (the tangent space
to ''M'' at ''p'') there exists a unique geodesic
: ''I'' → ''M'' such that
: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–Lindelöf theorem
for the solutions of ODEs with prescribed initial conditions. γ depends smoothly
on both ''p'' and ''V''.
In general, ''I'' may not be all of R as for example for an open disc in R2
. Any extends to all of if and only if is geodesically complete
is a local R-action
on the tangent bundle
''TM'' of a manifold ''M'' defined in the following way
where ''t'' ∈ R, ''V'' ∈ ''TM'' and
denotes the geodesic with initial data
. Thus, ''
''(''V'') = exp(''tV'') is the exponential map
of the vector ''tV''. A closed orbit of the geodesic flow corresponds to a closed geodesic
On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a Hamiltonian flow
on the cotangent bundle. The 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
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.
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
More precisely, an affine connection gives rise to a splitting of the double tangent bundle
TT''M'' into horizontal
and vertical bundle
The geodesic spray is the unique horizontal vector field ''W'' satisfying
at each point ''v'' ∈ T''M''; here π∗
: TT''M'' → T''M'' denotes the pushforward (differential)
along the projection π : T''M'' → ''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'' \ ) 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'' ∈ T''M'' \ and λ > 0. Here ''d''(''S''λ
) 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 () 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 () are called geodesics with affine parameter.
An affine connection is ''determined by'' its family of affinely parameterized geodesics, up to torsion
. 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
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
Efficient solvers for the minimal geodesic problem on surfaces posed as eikonal equation
s have been proposed by Kimmel and others.
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
* particle motion in molecular dynamics (MD) computer simulations
* robot motion planning
(e.g., when painting car parts); see Shortest path problem
* Differential geometry of surfaces
* Geodesic circle
*. ''See chapter 2''.
*. ''See section 2.7''.
*. ''See section 1.4''.
*. ''See section 87''.
*. Note especially pages 7 and 10.
*. ''See chapter 3''.
— 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).
Geodesics on a parametric surface -- sage interact
— Interactive SageMath
worksheet to calculate and illustrate geodesics on parametric surfaces.
Totally geodesic submanifold
at the Manifold Atlas