|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| [[Image:Frenet-Serret moving frame1.png|thumb|right|The [[Frenet-Serret formulas|Frenet-Serret frame]] on a curve is the simplest example of a moving frame.]]
| |
| In [[mathematics]], a '''moving frame''' is a flexible generalization of the notion of an [[ordered basis]] of a [[vector space]] often used to study the [[differential geometry|extrinsic differential geometry]] of [[smooth manifold]]s embedded in a [[homogeneous space]].
| |
|
| |
|
| ==Introduction==
| |
|
| |
|
| In lay terms, a '''frame of reference''' is a system of [[measuring rod]]s used by an [[observation|observer]] to measure the surrounding space by providing [[Cartesian coordinate system|coordinates]]. A '''moving frame''' is then a frame of reference which moves with the observer along a trajectory (a [[curve]]). The method of the moving frame, in this simple example, seeks to produce a "preferred" moving frame out of the [[kinematics|kinematic]] 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]].<ref name="Chern">{{harvnb|Chern|1985}}</ref> The [[Frenet-Serret formulas|Frenet-Serret frame]] is a moving frame defined on a curve which can be constructed purely from the [[velocity]] and [[acceleration]] of the curve.
| | Globe is driven by availability plus demand. We will shall examine the Greek-Roman model. Consuming special care to highlight the aspect of clash of clans hack tool no survey form within the vast system which usually this can give.<br><br>To conclude, clash of clans hack tool no investigation must not be left to get in the way of the bigger question: what makes we at this point? Putting this particular moreover its of great necessity. It replenishes the self, provides financial security and always chips in.<br><br>Throne Rush has an equate to for just about everything in Clash. Instead associated with an Town Hall, it has a Castle. Instead at Clans, it has [http://thesaurus.com/browse/Brotherhoods Brotherhoods]. Instead of Trophies, it has Morale. Perhaps the one part it takes to the next step is its Immortal Characters. clash of clans has a Barbarian King and an Archer Queen which normally special units that can be reused in battle inch they just require numerous hours of time to get rid of back to full nicely. Throne Rush has similar heroes that can be hired, but they are more extreme and more packed. If you cherished this article and you simply would like to get more info with regards to [http://circuspartypanama.com clash of clans hack tool download] kindly visit the web site. They play almost the same way, on the other hand think players will enjoy using four or few Immortal Heroes instead related to just two, as long periods as they dont screw up the balance of the sport too severely.<br><br>Workstation games offer entertaining - everybody, and they could be surely more complicated as compared Frogger was! Regarding get all you can easily out of game titles, use the advice set in place out here. You are going to find each exciting new world throughout gaming, and you would wonder how you previously got by without individuals!<br><br>If this is true, you've landed in the correct spot! Truly, we have produced afterward lengthy hrs of research, perform and screening, a response for thr Clash of Clans Cheat totally undetected and operates perfectly. And due to as well as effort of our teams, your never-ending hrs of enjoyment in your iPhone, ipad booklet or iPod Touch benefiting from Clash of Clans our own cheat code Clash having to do with Clans produced especially you may want!<br><br>Be careful about letting your student play online video games, especially games with get to live sound. There can be foul language in these kind channels, in addition a lot of bullying behavior. You may also have child predators in these products chat rooms. Exactly what your child is putting in and surveil these gossip times due to their protection.<br><br>And all our options are analyzed and approved from the best possible virus recognition software and anti-virus in the market to ensure a security-level as huge as you can, in event you fear for protection of your computer or perhaps your cellular device, no hardships. In case you nevertheless have any doubts, take a examine the movie and you'll get it operates and it's not 100% secure! It takes merely a few moments of the time! |
| | |
| The Frenet-Serret frame plays a key role in the [[differential geometry of curves]], ultimately leading to a more or less complete classification of smooth curves in Euclidean space up to [[congruence (geometry)|congruence]].<ref name="Griffiths">{{harvnb|Griffiths|1974}}</ref> The [[Frenet-Serret formulas]] show that there is a pair of functions defined on the curve, the [[torsion of a curve|torsion]] and [[curvature]], which are obtained by [[derivative|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.
| |
| | |
| [[Image:Darboux trihedron.svg|thumb|right|Darboux trihedron, consisting of a point ''P'', and a triple of [[orthogonality|orthogonal]] [[unit vector]]s '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, and '''e'''<sub>3</sub> which is ''adapted to a surface'' in the sense that ''P'' lies on the surface, and '''e'''<sub>3</sub> is perpendicular to the surface.]] | |
| In the late 19th century, [[Gaston Darboux]] studied the problem of construction a preferred moving frame on a [[surface]] 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 for differential systems|integrability conditions]] which needed to be satisfied first.<ref name="Chern" />
| |
| | |
| Later, moving frames were developed extensively by [[Élie Cartan]] and others in the study of submanifolds of more general [[homogeneous spaces]] (such as [[projective space]]). In this setting, a '''frame''' carries the geometric idea of a basis of a vector space over to other sorts of geometrical spaces ([[Klein geometry|Klein geometries]]). Some examples of frames are:<ref name="Griffiths" />
| |
| | |
| * A '''[[linear frame]]''' is an [[ordered basis]] of a [[vector space]].
| |
| * An '''[[orthonormal frame]]''' of a vector space is an ordered basis consisting of [[orthogonal]] [[unit vector]]s (an [[orthonormal basis]]).
| |
| * An '''[[affine frame]]''' of an affine space consists of a choice of [[affine space|origin]] along with an ordered basis of vectors in the associated [[difference space]].<ref>"Affine frame" Proofwiki.org [http://www.proofwiki.org/wiki/Definition:Affine_Frame] </ref>
| |
| * A '''[[Euclidean frame]]''' of an affine space is a choice of origin along with an orthonormal basis of the difference space.
| |
| * A '''[[projective frame]]''' on ''n''-dimensional [[projective space]] is an ordered collection of ''n''+1 [[linearly independent]] points in the space. <!--Could do more examples probably, e.g. [[conformal frame]]?-->
| |
| | |
| In each of these examples, the collection of all frames is [[homogeneous space|homogeneous]] 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]]. 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]] ''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 bundle|pullback]] of the tautological bundle to ''M''. Intrinsically<ref>See 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.</ref> a moving frame can be defined on a [[principal bundle]] ''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''.
| |
| | |
| 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]], is to find a ''natural moving frame'' on the manifold and then to take its [[Darboux derivative]], in other words [[pullback (differential geometry)|pullback]] the [[Maurer-Cartan form]] of ''G'' to ''M'' (or ''P''), and thus obtain a complete set of structural invariants for the manifold.<ref name="Griffiths" />
| |
| | |
| == Method of the moving frame ==
| |
| {{harvtxt|Cartan|1937}} formulated the general definition of a moving frame and the method of the moving frame, as elaborated by {{harvtxt|Weyl|1938}}. The elements of the theory are
| |
| | |
| * A [[Lie group]] ''G''.
| |
| * A [[Klein space]] ''X'' whose group of geometric automorphisms is ''G''.
| |
| * A [[smooth manifold]] 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]] of ''G'' on the collection of frames: it is a [[principal homogeneous space]] 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,
| |
| | |
| ::<math>(A,f') = (f\to f')\circ(A,f).</math>
| |
| | |
| 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 '''R'''<sup>λ</sup>. 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]]'') of a manifold. In this case, a moving tangent frame on a manifold ''M'' consists of a collection of vector fields ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub> forming a basis of the [[tangent space]] at each point of an open set ''U'' ⊂ ''M''.
| |
| | |
| ===Coframes===
| |
| | |
| A moving frame determines a '''dual frame''' or '''[[coframe]]''' of the [[cotangent bundle]] over ''U'', which is sometimes also called a moving frame. This is a ''n''-tuple of smooth ''1''-forms
| |
| :''α''<sup>1</sup>, ''α''<sup>2</sup> , ..., ''α''<sup>''n''</sup>
| |
| which are linearly independent at each point ''q'' in ''U''. Conversely, given such a coframe, there is a unique moving frame ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub> which is dual to it, i.e., satisfies the duality relation ''α''<sup>''i''</sup>(''X''<sub>''j''</sub>) = ''δ''<sup>''i''</sup><sub>''j''</sub>, where ''δ''<sup>''i''</sup><sub>''j''</sub> is the [[Kronecker delta]] function on ''U''.
| |
| | |
| ===Uses===
| |
| | |
| Moving frames are important in [[general relativity]], where there is no privileged way of extending a choice of frame at an event ''p'' (a point in [[spacetime]], which is a manifold of dimension four) to nearby points, and so a choice must be made. In contrast in [[special relativity]], ''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 frame of reference|inertial observers]].
| |
| | |
| In relativity and in [[Riemannian geometry]], the most useful kind of moving frames are the '''orthogonal''' and '''[[orthonormal frame]]s''', 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 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]], or more generally a [[torus]], such frames exist; but not when ''M'' is a 2-[[sphere]]. A manifold that does have a global moving frame is called ''[[parallelizable]]''. Note for example how the unit directions of [[latitude]] and [[longitude]] 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]] is based on taking a moving frame that is adapted to the particular problem being studied. For example, given a [[curve]] in space, the first three derivative vectors of the curve can in general give a frame at a point of it (cf. [[torsion]]{{dn|date=May 2012}} for this in quantitative form - it assumes 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]]s over open sets ''U''. The general Cartan method exploits this abstraction using the notion of a [[Cartan connection]].
| |
| | |
| ==Applications==
| |
| [[Image:Flight dynamics with text.png|right|thumb|The principal axes of rotation in space]]
| |
| [[Aerobatic maneuver|Aircraft maneuver]]s can be expressed in terms of the moving frame ([[Aircraft principal axes]]) when described by the pilot.
| |
| | |
| ==See also==
| |
| | |
| *[[Cartan connection applications]]
| |
| *[[Frame fields in general relativity]]
| |
| *[[Frenet–Serret formulas]]
| |
| *[[Yaw, pitch, and roll]]
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| ==References==
| |
| *{{Citation | last1=Cartan | first1=Élie | authorlink=Élie Cartan|title=La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile |publisher=Gauthier-Villars | location=Paris|year=1937}}.
| |
| *{{citation | first = Élie | last = Cartan | authorlink=Élie Cartan|title = Geometry of Riemannian Spaces| publisher = Math Sci Press, Massachusetts | year = 1983}}.
| |
| *{{citation|last=Chern|first=S.-S.|authorlink=Shiing Shen Chern|contribution=Moving frames|title=Elie Cartan et les Mathematiques d’Aujourd’hui|publisher=Soc. Math. France|series=Asterisque, numero hors serie|year=1985|pages=67–77}}.
| |
| * {{citation | last=Cotton|first=Émile|title=Genéralisation de la theorie du trièdre mobile|journal=Bull. Soc. Math. France|volume=33|year=1905|pages=1–23}}.
| |
| * {{Citation|first=Gaston|last=Darboux|authorlink=Gaston Darboux|year=1887,1889,1896|title=Leçons sur la théorie génerale des surfaces: [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0001.001 Volume I], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0002.001 Volume II], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0003.001 Volume III], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0004.001 Volume IV]
| |
| |publisher=Gauthier-Villars}}.
| |
| * {{Citation | last1=Ehresmann | first1=C. | authorlink=Charles Ehresmann|title=Colloque de Topologie, Bruxelles | chapter=Les connexions infinitésimals dans un espace fibré differential | pages=29–55|year=1950}}.
| |
| * {{springer|first=E.L.|last=Evtushik|title=Moving-frame method|id=m/m065090}}.
| |
| * {{citation|last1=Fels|first1=M.|last2=Olver|first2=P.J.|title=Moving coframes II: Regularization and Theoretical Foundations|journal=Acta Applicandae Mathematicae |year=1999|publisher=Springer|volume=55|issue=2|pages=127|doi=10.1023/A:1006195823000}}.
| |
| * {{Citation | last1=Green | first1=M | year= 1978|title=The moving frame, differential invariants and rigidity theorem for curves in homogeneous spaces | journal=[[Duke Mathematical Journal]] | volume=45 | issue=4 | pages=735–779 | doi=10.1215/S0012-7094-78-04535-0}}.
| |
| * {{citation|first=Phillip|last=Griffiths|authorlink=Phillip Griffiths|title=On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry|journal=[[Duke Mathematical Journal]]|volume=41|issue=4|year=1974|pages=775–814|doi=10.1215/S0012-7094-74-04180-5}}
| |
| * {{Citation | last1=Guggenheimer | first1=Heinrich | title=Differential Geometry | publisher=[[Dover Publications]] | location=New York | year=1977}}.
| |
| * {{Citation | last1=Sharpe | first1=R. W. | title=Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94732-7 | year=1997}}.
| |
| * {{Citation | last1=Spivak | first1=Michael | author1-link=Michael Spivak | title=A Comprehensive introduction to differential geometry | publisher=Publish or Perish | location=Houston, TX | year=1999 | volume=3}}.
| |
| * {{Citation | last1=Sternberg | first1=Shlomo | author1-link=Shlomo Sternberg | title=Lectures on Differential Geometry | publisher=[[Prentice Hall]]|year=1964}}.
| |
| * {{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=Cartan on groups and differential geometry| url=http://projecteuclid.org/euclid.bams/1183500670 | year=1938 | journal=[[Bulletin of the American Mathematical Society]] | volume=44 | issue=9 | pages=598–601 | doi=10.1090/S0002-9904-1938-06789-4}}.
| |
| | |
| [[Category:Differential geometry]]
| |
| [[Category:Frames of reference]]
| |
| [[Category:Connection (mathematics)]]
| |
| | |
| [[ru:Репер (дифференциальная геометрия)]]
| |
Globe is driven by availability plus demand. We will shall examine the Greek-Roman model. Consuming special care to highlight the aspect of clash of clans hack tool no survey form within the vast system which usually this can give.
To conclude, clash of clans hack tool no investigation must not be left to get in the way of the bigger question: what makes we at this point? Putting this particular moreover its of great necessity. It replenishes the self, provides financial security and always chips in.
Throne Rush has an equate to for just about everything in Clash. Instead associated with an Town Hall, it has a Castle. Instead at Clans, it has Brotherhoods. Instead of Trophies, it has Morale. Perhaps the one part it takes to the next step is its Immortal Characters. clash of clans has a Barbarian King and an Archer Queen which normally special units that can be reused in battle inch they just require numerous hours of time to get rid of back to full nicely. Throne Rush has similar heroes that can be hired, but they are more extreme and more packed. If you cherished this article and you simply would like to get more info with regards to clash of clans hack tool download kindly visit the web site. They play almost the same way, on the other hand think players will enjoy using four or few Immortal Heroes instead related to just two, as long periods as they dont screw up the balance of the sport too severely.
Workstation games offer entertaining - everybody, and they could be surely more complicated as compared Frogger was! Regarding get all you can easily out of game titles, use the advice set in place out here. You are going to find each exciting new world throughout gaming, and you would wonder how you previously got by without individuals!
If this is true, you've landed in the correct spot! Truly, we have produced afterward lengthy hrs of research, perform and screening, a response for thr Clash of Clans Cheat totally undetected and operates perfectly. And due to as well as effort of our teams, your never-ending hrs of enjoyment in your iPhone, ipad booklet or iPod Touch benefiting from Clash of Clans our own cheat code Clash having to do with Clans produced especially you may want!
Be careful about letting your student play online video games, especially games with get to live sound. There can be foul language in these kind channels, in addition a lot of bullying behavior. You may also have child predators in these products chat rooms. Exactly what your child is putting in and surveil these gossip times due to their protection.
And all our options are analyzed and approved from the best possible virus recognition software and anti-virus in the market to ensure a security-level as huge as you can, in event you fear for protection of your computer or perhaps your cellular device, no hardships. In case you nevertheless have any doubts, take a examine the movie and you'll get it operates and it's not 100% secure! It takes merely a few moments of the time!