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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a moving frame is a flexible generalization of the notion of an
ordered basis In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
often used to study the extrinsic differential geometry of
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s embedded in a
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements ...
.


Introduction

In lay terms, a ''
frame of reference In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
'' is a system of
measuring rod A measuring rod is a tool used to physically measure lengths and survey areas of various sizes. Most measuring rods are round or square sectioned; however, they can also be flat boards. Some have markings at regular intervals. It is likely th ...
s used by an
observer An observer is one who engages in observation or in watching an experiment. Observer may also refer to: Computer science and information theory * In information theory, any system which receives information from an object * State observer in co ...
to measure the surrounding space by providing
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
. A moving frame is then a frame of reference which moves with the observer along a
trajectory A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tr ...
(a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
). The method of the moving frame, in this simple example, seeks to produce a "preferred" moving frame out of the
kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
properties of the observer. In a geometrical setting, this problem was solved in the mid 19th century by Jean Frédéric Frenet and
Joseph Alfred Serret Joseph Alfred Serret (; August 30, 1819 – March 2, 1885) was a French people, French mathematician who was born in Paris, France, and died in Versailles (city), Versailles, France. See also *Frenet–Serret formulas Books by J. A. Serret Trai ...
. The Frenet–Serret frame is a moving frame defined on a curve which can be constructed purely from the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
and
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
of the curve. The Frenet–Serret frame plays a key role in the
differential geometry of curves Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the ...
, ultimately leading to a more or less complete classification of smooth curves in Euclidean space up to congruence. The
Frenet–Serret formulas In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \mathbb^, or the geometric properties of the curve itself irrespective ...
show that there is a pair of functions defined on the curve, the torsion and
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
, which are obtained by differentiating the frame, and which describe completely how the frame evolves in time along the curve. A key feature of the general method is that a preferred moving frame, provided it can be found, gives a complete kinematic description of the curve. In the late 19th century, Gaston Darboux studied the problem of constructing a preferred moving frame on a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
in Euclidean space instead of a curve, the Darboux frame (or the ''trièdre mobile'' as it was then called). It turned out to be impossible in general to construct such a frame, and that there were integrability conditions which needed to be satisfied first. Later, moving frames were developed extensively by
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
and others in the study of submanifolds of more general
homogeneous spaces In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ''G ...
(such as
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
). In this setting, a frame carries the geometric idea of a basis of a vector space over to other sorts of geometrical spaces ( Klein geometries). Some examples of frames are: * A linear frame is an
ordered basis In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. * An orthonormal frame of a vector space is an ordered basis consisting of
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
s (an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
). * An affine frame of an affine space consists of a choice of
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
along with an ordered basis of vectors in the associated difference space. * A
Euclidean frame Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of: Geometry * Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry ...
of an affine space is a choice of origin along with an orthonormal basis of the difference space. * A
projective frame In mathematics, and more specifically in projective geometry, a projective frame or projective basis is a tuple of points in a projective space that can be used for defining homogeneous coordinates in this space. More precisely, in a projectiv ...
on ''n''-dimensional
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
is an ordered collection of ''n''+1
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
points in the space. *
Frame fields in general relativity A frame field in general relativity (also called a tetrad or vierbein) is a set of four pointwise- orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of space ...
are four-dimensional frames, or
vierbein The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independen ...
s, in German. In each of these examples, the collection of all frames is
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
in a certain sense. In the case of linear frames, for instance, any two frames are related by an element of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
. Projective frames are related by the projective linear group. This homogeneity, or symmetry, of the class of frames captures the geometrical features of the linear, affine, Euclidean, or projective landscape. A moving frame, in these circumstances, is just that: a frame which varies from point to point. Formally, a frame on a
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements ...
''G''/''H'' consists of a point in the tautological bundle ''G'' → ''G''/''H''. A ''moving frame'' is a section of this bundle. It is ''moving'' in the sense that as the point of the base varies, the frame in the fibre changes by an element of the symmetry group ''G''. A moving frame on a submanifold ''M'' of ''G''/''H'' is a section of the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
of the tautological bundle to ''M''. IntrinsicallySee Cartan (1983) 9.I; Appendix 2 (by Hermann) for the bundle of tangent frames. Fels and Olver (1998) for the case of more general fibrations. Griffiths (1974) for the case of frames on the tautological principal bundle of a homogeneous space. a moving frame can be defined on a
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
''P'' over a manifold. In this case, a moving frame is given by a ''G''-equivariant mapping φ : ''P'' → ''G'', thus ''framing'' the manifold by elements of the Lie group ''G''. One can extend the notion of frames to a more general case: one can "
solder Solder (; NA: ) is a fusible metal alloy used to create a permanent bond between metal workpieces. Solder is melted in order to wet the parts of the joint, where it adheres to and connects the pieces after cooling. Metals or alloys suitable ...
" a
fiber bundle 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 and a ...
to a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, in such a way that the fibers behave as if they were tangent. When the fiber bundle is a homogenous space, this reduces to the above-described frame-field. When the homogenous space is a quotient of
special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
s, this reduces to the standard conception of a
vierbein The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independen ...
. Although there is a substantial formal difference between extrinsic and intrinsic moving frames, they are both alike in the sense that a moving frame is always given by a mapping into ''G''. The strategy in Cartan's method of moving frames, as outlined briefly in
Cartan's equivalence method In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism. For example, if ''M'' and ''N'' are two Riemannian manifolds with metrics ' ...
, is to find a ''natural moving frame'' on the manifold and then to take its
Darboux derivative The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. It is arguably a more natural generalization of the single-variable derivative. It allows a generalization of the single-variable fundamental ...
, in other words
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
the Maurer-Cartan form of ''G'' to ''M'' (or ''P''), and thus obtain a complete set of structural invariants for the manifold.


Method of the moving frame

formulated the general definition of a moving frame and the method of the moving frame, as elaborated by . The elements of the theory are * A
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
''G''. * A
Klein space In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space ''X'' together with a transitive action on ''X'' by a Lie group ''G'', which acts ...
''X'' whose group of geometric automorphisms is ''G''. * A
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
Σ which serves as a space of (generalized) coordinates for ''X''. * A collection of ''frames'' ƒ each of which determines a coordinate function from ''X'' to Σ (the precise nature of the frame is left vague in the general axiomatization). The following axioms are then assumed to hold between these elements: * There is a free and transitive
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of ''G'' on the collection of frames: it is a
principal homogeneous space In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
for ''G''. In particular, for any pair of frames ƒ and ƒ′, there is a unique transition of frame (ƒ→ƒ′) in ''G'' determined by the requirement (ƒ→ƒ′)ƒ = ƒ′. * Given a frame ƒ and a point ''A'' ∈ ''X'', there is associated a point ''x'' = (''A'',ƒ) belonging to Σ. This mapping determined by the frame ƒ is a bijection from the points of ''X'' to those of Σ. This bijection is compatible with the law of composition of frames in the sense that the coordinate ''x''′ of the point ''A'' in a different frame ƒ′ arises from (''A'',ƒ) by application of the transformation (ƒ→ƒ′). That is, (A,f') = (f\to f')\circ(A,f). Of interest to the method are parameterized submanifolds of ''X''. The considerations are largely local, so the parameter domain is taken to be an open subset of Rλ. Slightly different techniques apply depending on whether one is interested in the submanifold along with its parameterization, or the submanifold up to reparameterization.


Moving tangent frames

The most commonly encountered case of a moving frame is for the bundle of tangent frames (also called the ''
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
'') of a manifold. In this case, a moving tangent frame on a manifold ''M'' consists of a collection of vector fields ''e''1, ''e''2, …, ''e''''n'' forming a basis of the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at each point of an open set . If (x^1,x^2,\dots,x^n) is a coordinate system on ''U'', then each vector field ''ej'' can be expressed as a linear combination of the coordinate vector fields \frac:e_j = \sum_^n A^i_j \frac,where each A^i_j is a function on ''U''. These can be seen as the components of a matrix A. This matrix is useful for finding the coordinate expression of the dual coframe, as explained in the next section.


Coframes

A moving frame determines a dual frame or
coframe In mathematics, a coframe or coframe field on a smooth manifold M is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of M, one has a natural map from v_k:\bigoplus^kT^*M\to\big ...
of the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. Th ...
over ''U'', which is sometimes also called a moving frame. This is a ''n''-tuple of smooth ''1''-forms :''θ''1, ''θ''2, …, ''θ''''n'' which are linearly independent at each point ''q'' in ''U''. Conversely, given such a coframe, there is a unique moving frame ''e''1, ''e''2, …, ''e''''n'' which is dual to it, i.e., satisfies the duality relation ''θ''''i''(''e''''j'') = ''δ''''i''''j'', where ''δ''''i''''j'' is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
function on ''U''. If (x^1,x^2,\dots,x^n) is a coordinate system on ''U'', as in the preceding section, then each covector field ''θ''i can be expressed as a linear combination of the coordinate covector fields dx^i:\theta^i = \sum_^n B^i_j dx^j,where each B^i_j is a function on ''U.'' Since dx^i \left(\frac\right) = \delta^i_j, the two coordinate expressions above combine to yield \sum_^n B^i_k A^k_j = \delta^i_j ; in terms of matrices, this just says that A and B are inverses of each other. In the setting 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 ...
, when working with
canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cl ...
, the canonical coframe is given by the tautological one-form. Intuitively, it relates the velocities of a mechanical system (given by vector fields on the tangent bundle of the coordinates) to the corresponding momenta of the system (given by vector fields in the cotangent bundle; i.e. given by forms). The tautological one-form is a special case of the more general
solder form In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitiv ...
, which provides a (co-)frame field on a general
fiber bundle 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 and a ...
.


Uses

Moving frames are important 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 ...
, where there is no privileged way of extending a choice of frame at an event ''p'' (a point 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 ...
, which is a manifold of dimension four) to nearby points, and so a choice must be made. In contrast in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
, ''M'' is taken to be a vector space ''V'' (of dimension four). In that case a frame at a point ''p'' can be translated from ''p'' to any other point ''q'' in a well-defined way. Broadly speaking, a moving frame corresponds to an observer, and the distinguished frames in special relativity represent inertial observers. In relativity and in
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, the most useful kind of moving frames are the orthogonal and orthonormal frames, that is, frames consisting of orthogonal (unit) vectors at each point. At a given point ''p'' a general frame may be made orthonormal by
orthonormalization In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors in an inner product space (most commonly the Euclidean s ...
; in fact this can be done smoothly, so that the existence of a moving frame implies the existence of a moving orthonormal frame.


Further details

A moving frame always exists ''locally'', i.e., in some neighbourhood ''U'' of any point ''p'' in ''M''; however, the existence of a moving frame globally on ''M'' requires topological conditions. For example when ''M'' is a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
, or more generally 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 ...
, such frames exist; but not when ''M'' is a 2-
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 ...
. A manifold that does have a global moving frame is called '' parallelizable''. Note for example how the unit directions of
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
and
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
on the Earth's surface break down as a moving frame at the north and south poles. The method of moving frames of
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
is based on taking a moving frame that is adapted to the particular problem being studied. For example, given a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
in space, the first three derivative vectors of the curve can in general define a frame at a point of it (cf.
torsion tensor In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a cur ...
for a quantitative description – it is assumed here that the torsion is not zero). In fact, in the method of moving frames, one more often works with coframes rather than frames. More generally, moving frames may be viewed as sections of
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
s over open sets ''U''. The general Cartan method exploits this abstraction using the notion of a
Cartan connection In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the ...
.


Atlases

In many cases, it is impossible to define a single frame of reference that is valid globally. To overcome this, frames are commonly pieced together to form an
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
, thus arriving at the notion of a local frame. In addition, it is often desirable to endow these atlases with a
smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. Definition A smooth structure on a manifold M is ...
, so that the resulting frame fields are differentiable.


Generalizations

Although this article constructs the frame fields as a coordinate system on the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, the general ideas move over easily to the concept of a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
, which is a manifold endowed with a vector space at each point, that vector space being arbitrary, and not in general related to the tangent bundle.


Applications

Aircraft maneuvers can be expressed in terms of the moving frame (
Aircraft principal axes An aircraft in flight is free to rotate in three dimensions: '' yaw'', nose left or right about an axis running up and down; ''pitch'', nose up or down about an axis running from wing to wing; and ''roll'', rotation about an axis running from ...
) when described by the pilot.


See also

* Darboux frame *
Frenet–Serret formulas In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \mathbb^, or the geometric properties of the curve itself irrespective ...
* Yaw, pitch, and roll


Notes


References

*. *. *. * . * . * . * . * . * . * . * . * . * * . * . * . * . * . {{Manifolds Connection (mathematics) Differential geometry Frames of reference ru:Репер (дифференциальная геометрия)