In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a geodesic () is 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".
The noun ''
geodesic'' and the adjective ''
geodetic'' come from ''
geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
'', the science of measuring the size and shape of
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
, though 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
Spherical Earth or Earth's curvature refers to the approximation of figure of the Earth as a sphere.
The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of Greek philosophers. ...
, it is a
segment of a
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
(see also
great-circle distance
The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle.
It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a st ...
). The term has since been generalized to more abstract mathematical spaces; for example, in
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, 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
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
s remain parallel if they are
transported along it. Applying this to the
Levi-Civita connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
of a
Riemannian metric recovers the previous notion.
Geodesics are of particular importance in
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. Timelike
geodesics in general relativity describe the motion of
free falling
test particles In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be ins ...
.
Introduction
A locally shortest path between two given points in a curved space, assumed to be a
Riemannian manifold, can be defined by using the
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...
for the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
of a
curve (a function ''f'' from an
open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
of
R to the space), and then minimizing this length between the points using the
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. 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 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
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
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
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, geodesics in
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
describe the motion of
point particles under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting
satellite
A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioiso ...
, 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 manifolds. The article
Levi-Civita connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
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
The most familiar examples are the straight lines in
Euclidean geometry. On a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
, the images of geodesics are the
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
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 points, 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
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
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
arcs, forming a
spherical triangle
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...
.
Metric geometry
In
metric geometry, a geodesic is a curve which is everywhere
locally a
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
minimizer. More precisely, a
curve from an interval ''I'' of the reals to the
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
''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 In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second alon ...
are joined by a minimizing sequence of
rectifiable path
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
s, although this minimizing sequence need not converge to a geodesic.
Riemannian geometry
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 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
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
.
Calculus of variations
Techniques of the classical
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
can be applied to examine the energy functional ''E''. The
first variation In applied mathematics and the calculus of variations, the first variation of a functional ''J''(''y'') is defined as the linear functional \delta J(y) mapping the function ''h'' to
:\delta J(y,h) = \lim_ \frac = \left.\frac J(y + \varepsilon h ...
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
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, one can also regard
geodesics as Hamiltonian flows In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set of coupled first-order equation ...
. They are solutions of the associated
Hamilton equations, with (pseudo-)Riemannian metric taken as
Hamiltonian.
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
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
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
. 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) as
:
where
are the coordinates of the curve γ(''t'') and
are the
Christoffel symbols of the connection ∇. This is an
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
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
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, geodesics can be thought of as trajectories of
free particles 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 ''T
pM'' (the
tangent space to ''M'' at ''p'') there exists a unique geodesic
: ''I'' → ''M'' such that
::
and
::
:where ''I'' is a maximal
open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
in R containing 0.
The proof of this theorem follows from the theory of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
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 R
2. Any extends to all of if and only if is
geodesically complete In mathematics, a complete manifold (or geodesically complete manifold) is a ( pseudo-) Riemannian manifold for which, starting at any point , you can follow a "straight" line indefinitely along any direction. More formally, the exponential map ...
.
Geodesic flow
Geodesic
flow
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
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 ''M''.
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
, i.e.
:
In particular, when ''V'' is a unit vector,
remains unit speed throughout, so the geodesic flow is tangent to the
unit tangent bundle
In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (''M'', ''g''), denoted by T1''M'', UT(''M'') or simply UT''M'', is the unit sphere bundle for the tangent bundle T(''M''). It is a fiber bundle over ''M'' whose fiber at each ...
.
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
tangent bundle. The derivatives of these curves define a
vector field on the
total space
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E an ...
of the tangent bundle, known as the geodesic
spray
Spray or spraying commonly refer to:
* Spray (liquid drop)
** Aerosol spray
** Blood spray
** Hair spray
** Nasal spray
** Pepper spray
** PAVA spray
** Road spray or tire spray, road debris kicked up from a vehicle tire
** Sea spray, refers to ...
.
More precisely, an affine connection gives rise to a splitting of the
double tangent bundle In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle of the total space ''TM'' of the tangent bundle of a smooth manifold ''M''
. A note on notation: in this ar ...
TT''M'' into
horizontal and
vertical bundles:
:
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
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it do ...
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
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curve ...
.
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
:
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 posed as
eikonal equation
An eikonal equation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation.
The classical eikonal equation in geometric optics is a differential equation o ...
s have been proposed by Kimmel and others.
Ribbon Test
A Ribbon "Test" is a way of finding a geodesic on a physical surface. 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
neighborhood of a line
in a plane into a surface
so that the mapping
"doesn't change the distances around
by much"; that is, at the distance
from
we have
where
and
are
metrics on
and
.
Applications
Geodesics serve as the basis to calculate:
* geodesic airframes; see
geodesic airframe or
geodetic airframe
* geodesic structures – for example
geodesic domes
A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the structural stress throughout the structure, making geodesic ...
* 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
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
The problem of finding the shortest path between ...
See also
*
*
*
*
Differential geometry of surfaces
*
Geodesic circle A geodesic circle is either "the locus on a surface at a constant geodesic distance from a fixed point" or a curve of constant geodesic curvature.
A geodesic disk is the region on a surface bounded by a geodesic circle.
In contrast with the ord ...
*
*
*
*
*
*
*
Notes
References
*
Further reading
*. ''See chapter 2''.
*. ''See section 2.7''.
*. ''See section 1.4''.
*.
*. ''See section 87''.
*
*. Note especially pages 7 and 10.
*.
*. ''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
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
), mechanics (
brachistochrone) and optics (light beam in inhomogeneous medium).
Totally geodesic submanifoldat the Manifold Atlas
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Differential geometry