From Frenet to Cartan: the Method of Moving Frames
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Moving Parseval Frames for Vector Bundles
Houston Journal of Mathematics c 2014 University of Houston Volume 40, No. 3, 2014 MOVING PARSEVAL FRAMES FOR VECTOR BUNDLES D. FREEMAN, D. POORE, A. R. WEI, AND M. WYSE Communicated by Vern I. Paulsen Abstract. Parseval frames can be thought of as redundant or linearly de- pendent coordinate systems for Hilbert spaces, and have important appli- cations in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for a vector bundle over a manifold. Many vector bundles do not have a moving basis, but in contrast to this every vector bundle over a paracompact manifold has a moving Parseval frame. We prove that a sequence of sections of a vector bundle is a moving Parseval frame if and only if the sections are the orthogonal projection of a moving orthonormal basis for a larger vector bundle. In the case that our vector bundle is the tangent bundle of a Riemannian manifold, we prove that a sequence of vector fields is a Parseval frame for the tangent bundle of a Riemannian manifold if and only if the vector fields are the orthogonal projection of a moving orthonormal basis for the tangent bundle of a larger Riemannian manifold. 1. Introduction Frames for Hilbert spaces are essentially redundant coordinate systems. That is, every vector can be represented as a series of scaled frame vectors, but the series is not unique. Though this redundancy is not necessary in a coordinate system, it can actually be very useful. -
Chapter 2: Minkowski Spacetime
Chapter 2 Minkowski spacetime 2.1 Events An event is some occurrence which takes place at some instant in time at some particular point in space. Your birth was an event. JFK's assassination was an event. Each downbeat of a butterfly’s wingtip is an event. Every collision between air molecules is an event. Snap your fingers right now | that was an event. The set of all possible events is called spacetime. A point particle, or any stable object of negligible size, will follow some trajectory through spacetime which is called the worldline of the object. The set of all spacetime trajectories of the points comprising an extended object will fill some region of spacetime which is called the worldvolume of the object. 2.2 Reference frames w 1 w 2 w 3 w 4 To label points in space, it is convenient to introduce spatial coordinates so that every point is uniquely associ- ated with some triplet of numbers (x1; x2; x3). Similarly, to label events in spacetime, it is convenient to introduce spacetime coordinates so that every event is uniquely t associated with a set of four numbers. The resulting spacetime coordinate system is called a reference frame . Particularly convenient are inertial reference frames, in which coordinates have the form (t; x1; x2; x3) (where the superscripts here are coordinate labels, not powers). The set of events in which x1, x2, and x3 have arbi- x 1 trary fixed (real) values while t ranges from −∞ to +1 represent the worldline of a particle, or hypothetical ob- x 2 server, which is subject to no external forces and is at Figure 2.1: An inertial reference frame. -
The Light Cone and the Conformal Sphere: Differential Invariants And
Light cone and conformal sphere The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Theresa C. Anderson April, 2010 Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere Motivation: The recent concept of the group-based moving frame as an equivariant map, introduced by Fels and Olver, has been a useful aid in the quest for differential invariants. The moving frame is used to construct the Maurer-Cartan matrix K; which according to a theorem of Hubert, contains all generators for the differential invariants of a curve. Here we employ the normalization technique of Fels and Olver to construct group-based moving frames for the light cone and the conformal 2-sphere, and we use the moving frame to find the differential invariants. Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Euclidean space admits a natural action from the Euclidean group, the group of rigid transformations (rotations and translations). Light cone and conformal sphere We first recall the following definition and theorem: A classical moving frame on a n-manifold is a set of n vectors along a curve which is invariant under an action by the group (such as a rigid motion in the Euclidean case). Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere We first recall the following definition and theorem: A classical moving frame on a n-manifold is a set of n vectors along a curve which is invariant under an action by the group (such as a rigid motion in the Euclidean case). -
Combining Points and Tangents Into Parabolic Polygons: an Affine
Combining points and tangents into parabolic polygons: an affine invariant model for plane curves MARCOS CRAIZER1,THOMAS LEWINER1 AND JEAN-MARIE MORVAN2 1 Department of Mathematics — Pontif´ıcia Universidade Catolica´ — Rio de Janeiro — Brazil 2 Universite´ Claude Bernard — Lyon — France {craizer, tomlew}@mat.puc--rio.br. [email protected]. Abstract. Image and geometry processing applications estimate the local geometry of objects using information localized at points. They usually consider information about the tangents as a side product of the points coordinates. This work proposes parabolic polygons as a model for discrete curves, which intrinsically combines points and tangents. This model is naturally affine invariant, which makes it particularly adapted to computer vision applications. As a direct application of this affine invariance, this paper introduces an affine curvature estimator that has a great potential to improve computer vision tasks such as matching and registering. As a proof–of–concept, this work also proposes an affine invariant curve reconstruction from point and tangent data. Keywords: Affine Differential Geometry. Affine Curvature. Affine Length. Curve Reconstruction. Figure 1: Parabolic polygon (right) obtained from a Lissajous curve with 10 samples vs. straight line polygon ignoring tangents (left). 1 Introduction measures. These normals can also be robustly estimated only from the point coordinates [11, 9], or from direct image pro- Computers represent geometric objects through discrete cessing [5, 16]. structures. These structures usually rely on point-wise in- formation combined with adjacency relations. In particular, However, the normal or tangent information is usually most geometry processing applications require the normal of considered separately from the point coordinate, and the def- the object at each point: either for rendering [15], deforma- inition of geometrical objects such as contour curves or dis- tion [4], or numerical stability of reconstruction [1]. -
Moving Frames
Moving Frames Peter J. Olver University of Minnesota http://www.math.umn.edu/ ∼ olver Benasque, September 2009 History of Moving Frames Classical contributions: M. Bartels (∼1800), J. Serret, J. Fr´enet, G. Darboux, ´ E. Cotton, El´ ie Cartan Modern developments: (1970’s) S.S. Chern, M. Green, P. Griffiths, G. Jensen, T. Ivey, J. Landsberg, . The equivariant approach: (1997 – ) PJO, M. Fels, G. Mar´ı–Beffa, I. Kogan, J. Cheh, J. Pohjanpelto, P. Kim, M. Boutin, D. Lewis, E. Mansfield, E. Hubert, E. Shemyakova, O. Morozov, R. McLenaghan, R. Smirnov, J. Yue, A. Nikitin, J. Patera, P. Vassiliou, . Moving Frame — Space Curves t b tangent normal binormal dz d2z n t = n = b = t × n ds ds2 s — arc length Fr´enet–Serret equations dt dn db = κ n = − κ t + τ b = − τ n ds ds ds κ — curvature τ — torsion Moving Frame — Space Curves t b tangent normal binormal dz d2z n t = n = b = t × n ds ds2 s — arc length z Fr´enet–Serret equations dt dn db = κ n = − κ t + τ b = − τ n ds ds ds κ — curvature τ — torsion “I did not quite understand how he [Cartan] does this in general, though in the examples he gives the procedure is clear.” “Nevertheless, I must admit I found the book, like most of Cartan’s papers, hard reading.” — Hermann Weyl “Cartan on groups and differential geometry” Bull. Amer. Math. Soc. 44 (1938) 598–601 Applications of Moving Frames • Differential geometry • Equivalence • Symmetry • Differential invariants • Rigidity • Identities and syzygies • Joint invariants and semi-differential invariants • Invariant differential forms and tensors • Integral invariants • Classical invariant theory • Computer vision ◦ object recognition ◦ symmetry detection • Invariant variational problems • Invariant numerical methods • Mechanics, including DNA • Poisson geometry & solitons • Killing tensors in relativity • Invariants of Lie algebras in quantum mechanics • Control theory • Lie pseudo-groups The Basic Equivalence Problem M — smooth m-dimensional manifold. -
Centro–Affine Curvature Flows on Centrally Symmetric Convex Curves
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 11, November 2014, Pages 5671–5692 S 0002-9947(2014)05928-X Article electronically published on July 21, 2014 CENTRO–AFFINE CURVATURE FLOWS ON CENTRALLY SYMMETRIC CONVEX CURVES MOHAMMAD N. IVAKI Abstract. We consider two types of p-centro-affine flows on smooth, cen- trally symmetric, closed convex planar curves: p-contracting and p-expanding. Here p is an arbitrary real number greater than 1. We show that, under any p-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the ori- gin. Furthermore, the normalized curves with enclosed area π converge, in the Hausdorff metric, to the unit circle modulo SL(2). As a p-expanding flow is, in a certain way, dual to a contracting one, we prove that, under any p-expanding flow, curves expand to infinity in finite time, while the only homothetic solu- tions of the flow are ellipses centered at the origin. If the curves are normalized to enclose constant area π, they display the same asymptotic behavior as the first type flow and converge, in the Hausdorff metric, and up to SL(2) trans- formations, to the unit circle. At the end of the paper, we present a new proof of the p-affine isoperimetric inequality, p ≥ 1, for smooth, centrally symmetric convex bodies in R2. 1. Introduction The affine normal flow is a widely recognized evolution equation for hypersurfaces in which each point moves with velocity given by the affine normal vector. -
Moving Frames Astérisque, Tome S131 (1985), P
Astérisque SHIING-SHEN CHERN Moving frames Astérisque, tome S131 (1985), p. 67-77 <http://www.numdam.org/item?id=AST_1985__S131__67_0> © Société mathématique de France, 1985, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Société Mathématique de France Astérisque, hors série, 1985, p. 67-77 MOVING FRAMES BY Shiing-shen CHERN1 Introduction The method of moving frames has a long history. It is at the heart of kinematics and its conscious application to differential geometry could be traced back at least to SERRET and FRENET2. In his famous "Leçons sur la théorie des surfaces," based on lectures at the Sorbonne 1882-1885, DAR- BOUX had the revolutionary idea using frames depending on two parameters and integrability conditions turn up. DARBOUX'S method was generalized to arbitrary Lie groups by Emile COTTON [14] for the search of differen tial invariants, whose work should be considered a forerunner of the general method of moving frames. CARTAN'S first paper on moving frames was published in 1910 [7]. He im mediately observed that DARBOUX'S partial derivatives should be combined into the "Maurer-Cartan forms" and DARBOUX'S integrability conditions are essentially the Maurer-Cartan equations. This extends the method to the case where the ambient space is acted on by any Lie group and, more generally, by an infinite pseudogroup in CARTAN'S sense, such as the pseu- dogroup of all complex analytic automorphisms in n variables. -
Modern Developments in the Theory and Applications of Moving Frames
London Mathematical Society Impact150 Stories 1 (2015) 14{50 C 2015 Author(s) doi:10.1112/l150lms/l.0001 e Modern Developments in the Theory and Applications of Moving Frames Peter J. Olver Abstract This article surveys recent advances in the equivariant approach to the method of moving frames, concentrating on finite-dimensional Lie group actions. A sampling from the many current applications | to geometry, invariant theory, and image processing | will be presented. 1. Introduction. According to Akivis, [2], the method of rep`eresmobiles, which was translated into English as moving framesy, can be traced back to the moving trihedrons introduced by the Estonian mathematician Martin Bartels (1769{1836), a teacher of both Gauß and Lobachevsky. The apotheosis of the classical development can be found in the seminal advances of Elie´ Cartan, [25, 26], who forged earlier contributions by Frenet, Serret, Darboux, Cotton, and others into a powerful tool for analyzing the geometric properties of submanifolds and their invariants under the action of transformation groups. An excellent English language treatment of the Cartan approach can be found in the book by Guggenheimer, [49]. The 1970's saw the first attempts, cf. [29, 45, 46, 64], to place Cartan's constructions on a firm theoretical foundation. However, the method remained constrained within classical geometries and homogeneous spaces, e.g. Euclidean, equi-affine, or projective, [35]. In the late 1990's, I began to investigate how moving frames and all their remarkable consequences might be adapted to more general, non-geometrically-based group actions that arise in a broad range of applications. -
Moving Frame Equations in Three Dimensional General Inner Product Space
Archive of SID Moving Frame Equations in Three Dimensional General Inner Product Space Ali Parsian Department of Mathematics, Tafresh University, Tafresh, Iran. [email protected] Abstract In this paper, we are going to generalize the Frenet-Serret formulas for the moving frames in three dimensional space 푅3, in the case that the space admits the general form of inner product. Keywords: Covariant derivative, Moving frame, Vector field. www.SID.ir Archive of SID Preliminaries Vectors are used widely in physics and engineering to describe forces, velocities, angular momentum, and many other concepts. To obtain a definition that is both practical and precise, we shall describe an “arrow” in 푅3 by giving its starting point 푝 and the change, or vector 푣, necessary to reach its end point 푝 + 푣. Strictly speaking, 푣 is just a point of 푅3. A tangent 3 3 vector 푣푝 to 푅 consists of two points of 푅 : its vector part 푣 and its point of application 푝. A vector field 푉 on 푅3 is a map that assigns to each point 푝 of 푅3 a tangent vector 푉(푝) to 푅3 at 푝. There is a natural algebra of vector fields. At each point 푝, the values 푉(푝) and 푊(푝) are 3 in the same vector space, the tangent space 푇푝푅 , consequently, the formula for the addition is thus the same as for addition of maps, (푉 + 푊)(푝) = 푉(푝) + 푊(푝) or all 푝 ∈ 푅3. If 푓 is a real-valued map on 푅3 and 푉 is a vector field on 푅3, then 푓푉 is defined to be the vector field on 푅3 such that (푓푉)(푝) = 푓(푝)푉(푝) for all 푝 ∈ 푅3. -
The Light Sphere Paradox of Special Relatively
The Light Sphere Paradox Of Special Relatively By Harry H. Ricker III Email: [email protected] 1.0 Introduction This paper studies the light sphere paradox of Einstein's special theory of relativity and demonstrates that the failure to establish a correct mathematical analysis of this problem has allowed a major flaw in that theory to be turned into the foundational mathematical basis of that theory, which is unfortunately false and erroneous. 2.0 Background Unlike the much more famous paradox of the twins, the light sphere paradox is relatively unknown. It has not produced the volume of critical discussion or commanded the attention of critics of relativity that it deserves. The author could find only one reference to this paradox in an Internet search and only four references to it in his collection of relativity books. 2.1 Description Of Paradox Of The Spheres The best introductory discussion of the paradox is presented in Basic Concepts of Relativity, by R. H. Good, pages 39 and 40. Good asks the reader “Suppose a light pulse emitted from a stationary source at the origin at time t=0. It then spreads out with velocity c in a spherical wave front, as seen by an observer at rest relative to the source...Now an observer in another frame of reference moving uniformly in the x direction with velocity v relative to the first frame will also see a spherical wave front spreading out with the velocity c from the origin of his reference frame, assuming that the origins coincide at t=0. -
Curvature Functionals for Curves in the Equi-Affine Plane 3
CURVATURE FUNCTIONALS FOR CURVES IN THE EQUI-AFFINE PLANE. STEVEN VERPOORT (K.U.Leuven, Belgium, and Masaryk University / Eduard Cechˇ Center, Czech Republic.) Abstract. After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail. Key Words. Curvature Functionals, Variational Problems, Affine Curves. AMS 2010 Classification. 49K05, 49K15, 53A15. 1. Preface.1 1 One of the many striking features of W. Blaschke's landmark book \Vorlesungen II " [4], being the first treatise on equi-affine differential geometry which also at present day remains in multiple aspects the best introduction to the subject, is the close analogy between the development of the main body of the equi-affine theory and the exposition of classical differential geometry [3]. Although Blaschke showed a great interest in isoperimetric and variational problems, a rare topic which breaks this similarity is precisely the question of the infinitesimal change of a planar curvature functional, for Radon's problem is indeed covered with some detail in [3] whereas in [4] only the variation of the equi-affine arc-length is considered. In fact, even after having asked many colleagues, to whom I extend my gratitude, I have not been able to find any article on equi-affine curvature functionals for planar curves (although centro-affine curvature functionals for planar curves are treated in [11] whereas [17] covers a variational problem w.r.t. the full-affine group). -
Parabolic Polygons and Discrete Affine Geometry
Parabolic Polygons and Discrete Affine Geometry Marcos Craizer Thomas Lewiner Jean-Marie Morvan Department of Mathematics, PUC–Rio, Brazil Universite´ Claude Bernard — Lyon — France Figure 1. Example of a parabolic polygon with 10 arcs (left), our estimation of their affine length (middle) and affine curvature (right). Abstract definition of the geometrical object depends rather on the point coordinates. Although modelling already makes in- Geometry processing applications estimate the local ge- tensive use of this information, in particular with Bezier´ ometry of objects using information localized at points. curves, only recent developments in reconstruction prob- They usually consider information about the normal as a lems proposed to incorporate these tangents as part of the side product of the points coordinates. This work proposes point set definition [10]. parabolic polygons as a model for discrete curves, which in- In this work, we propose a discrete curve representation trinsically combines points and normals. This model is nat- based on points and tangents: the parabolic polygons, intro- urally affine invariant, which makes it particularly adapted duced in Section 2. This model is naturally invariant with to computer vision applications. This work introduces es- respect to affine transformations of the plane. As opposed to timators for affine length and curvature on this discrete implicit affine representations [12], our representation uses model and presents, as a proof–of–concept, an affine in- only local information. This makes it particularly adapted variant curve reconstruction. to computer vision applications, since two contours of the Keywords: Affine Differential Geometry, Affine Curvature, same planar object obtained from different perspectives are Affine Length, Curve Reconstruction.