DOCSLIB.ORG

The Affine Curve-Lengthening Flow $

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Affine Curve-Lengthening Flow $ J. reine angew. Math. 50 6 (1999), 43-83 Journal fur die reine und angewandte Mathematik © W a l t e r d e G r u y t e r Berlin · New York 1999 The affine curve-lengthening flow B y Ben Andrews1) at Canberra A b s t r a c t .The motion o f any smooth closed convex curve in the plane i n the d i r e c ¬ tion o f steepest increase o f its a f f i n e arc length can be continued smoothly f o r all time. The evolving curve remains strictly convex w h i l e expanding to i n f i n i t e s i z e and approaching a homothetically expanding ellipse. 1. Introduction In this paper w e study an affine-geometric, fourth-order parabolic evolution equation f o r closed convex curves in the plane. This i s d e f i n e d by following the direction o f steepest ascent o f the a f f i n e arc length functional L, with respect to a natural affine-invariant inner product. The definitions o f these are g i v e n i n Section 2, along with a discussion o f other aspects o f the a f f i n e differential geometry o f curves. The evolution equation can be written as follows: ( 1 ) $ where $ is the affine normal vector, and $ is the affine curvature. In terms of Euclidean- geometric invariants, this can be written as follows: $ where is the Euclidean curvature of the curve, u is an anticlockwise Euclidean arc-length parameter, n is the outward unit normal, and t is the Euclidean unit tangent vector. This is a fifth-order system of partial differential equations which is equivalent to a scalar fourth- order parabolic equation (see Section 3). Equation (1) is the parabolic analogue of the affine-geometric maximal surface equation, which has been an important problem in affine differential geometry, and with regard to which many problems remain unsolved (see [NS] and [LSZ]). 1) Research partly supported by NSF grant DMS 9504456 and a Terman fellowship. Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 44 Andrews, The affine curve-lengthening flow The analogous Euclidean-geometric evolution equation, the curve-shortening flow, has been studied extensively b y Gage ([Ga1], [Ga2]), Gage and Hamilton [GH], Grayson [Gr] and many others. For that evolution equation i t has been shown that any initial closed, smooth embedded curve produces a solution which exists f o r a f i n i t e time, contract- ing at the end o f that time to a single point, in such a w a y that the curves can be rescaled about the final point to converge to a circle. The curve-shortening flow i s a second-order partial differential equation, and many details of its analysis are based on the parabolic maximum principle, which i s not available to us i n the present problem. However, the result w e prove i n this paper i s somewhat analogous to that f o r the curve-shortening flow: T h e o r e m 1 . Let $ be a smooth, strictly locally convex embedding of a curve $ into the plane. Then there exists a unique, smooth solution {x t } of th e ev o l u t i o n eq u a t i o n (1), which exists for all positive times. There exists a smooth embedding $, a point $, and a real number $, such that the image of x∞ is an ellipse of enclosed area π with centre at the origin, and $ Any e l l i p s e ev o l v e s b y expanding homothetically about its centre. Some other evolution equations o f higher order f o r curves i n the plane have been considered before - Polden [Po1] has considered the L2 gradient flow o f the integral o f the square o f the curvature, which i s a fourth-order parabolic equation (note that Langer and Singer [LS] earlier introduced a d i f f e r e n t gradient flow, i n which the integral o f the squared curvature i s treated as a functional on the space o f W2,1 indicatrices o f curves o f fixed length - this produces a non-local flow o f zeroth order; W e n [Wl-2] has also considered the gradient flow o f the same functional on the space o f L2 indicatrices o f curves o f fixed length, which i s a second order parabolic equation). Higher order equations have also been considered f o r the evolution o f Riemannian metrics, b y Rendall [Re], Singleton [Si], Chruściel [Ch], and Polden [Po2]. The main resuit used in characterising the long-term behaviour of solutions (presented in Section 4) is closely related to integral estimates which the author developed in appli¬ cations to second-order evolution equations for curves and for hypersurfaces ([A1]-[A3]). The regularity estimates for the flow, given here in Section 7, are inspired by the work of Alexander Polden [Po1-2] on higher order evolution equations for curves and for metrics on surfaces. The equation under investigation here has some remarkable special features not shared by most other higher-order equations, however: First, we have the advantage of a large group of symmetries (namely, the special affine group); second, the equation has a certain degeneracy (see Eq. (19)) which has important consequences for the analysis - although in many situations such degeneracy may considered a disadvantage, it is vital to the argument of the present paper, because non-degenerate fourth-order equations do not preserve pointwise inequalities, and in particular could not keep arbitrary convex curves convex. Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 45 The organisation o f the paper i s as follows: In Section 2 w e summarise various aspects o f the a f f i n e differential geometry o f curves, support functions and their use i n describing convex sets, and other results required i n the later analysis. In Section 3 w e introduce the evolution equation, establish the short-time existence o f solutions, and discuss the n o r m a ¬ lisation o f solution curves using scalings, translations, and area-preserving a f f i n e boosts. Section 4 accomplishes the k e y step i n the proof, using the Brunn-Minkowski theorem about concavity of the area functional to deduce an integral estimate with two important consequences: First, it implies that solutions which e x i s t f o r a l l time must evolve purely b y scaling i n the large-time limit; and secondly i t implies that the evolving curve remains strictly convex as long as it exists, as w e demonstrate i n Section 5 . Section 6 completes our characterisation o f the long-term behaviour b y showing that the only solutions which e v o l v e purely b y scaling are ellipses. This i s proved b y partially integrating the fourth-order ordinary differential equation characterising such solutions, and deriving a contradiction to the a f f i n e isoperimetric inequality i n a l l cases except ellipses. In particular w e make use o f an affine-geometric version o f the four-vertex theorem (Proposition 1 8 ) . Section 7 provides regularity estimates which imply that solutions e x i s t f o r a l l time, and Section 8 combines all these elements to complete the proof o f Theorem 1 . W e note that the classification o f homothetic solutions f o r this evolution equation has also been carried out b y Lima and Montenegro [LM], and their analysis also g i v e s a classification o f non-embedded homothetic solutions. A d i f f e r e n t affine-geometric evolution equation, the f l o w b y the a f f i n e normal vector, has been studied b y Tannenbaum and Sapiro [TS1-2] and the author [A4]. That evolution equation i s o f second order, and the methods associated with it are more c l o s e l y related to those applied to the curve- shortening f l o w . The author would like to thank Katsumi Nomizu f o r suggesting the equation con- sidered here, and Alexander Polden f o r some f r u i t f u l discussions concerning the regularity o f solutions. 2. Background a n d notation 2.1. Support functions. Let $ be a closed, smooth, convex embedded curve in the plane, and Ω the region enclosed by $. Th e su p p o r t fu n c t i o n $ o f $ ( o r Ω) is d e f i n e d b y ( 2 ) $. The support function contains all geometric information about the curve. In particular, there is a natural embedding $ with image equal to $, defined by (3) $ for all $, where we think of S 1 as na t u r a l l y in c l u d e d in $, an d θ is the usual angle coordinate on S1.
Load more

Recommended publications
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • Generic Equi-Centro-Affine Differential Geometry of Plane Curves
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 159 (2012) 476–483 Contents lists available at SciVerse ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Generic equi-centro-affine differential geometry of plane curves ∗ Peter J. Giblin a, , Takashi Sano b a Department of Mathematical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK b Faculty of Engineering, Hokkai-Gakuen University, Sapporo, 062-8605, Japan article info abstract MSC: We study the equi-centro-affine invariants of plane curves from the view point of the 53A15 singularity theory of smooth functions. We define the notion of the equi-centro-affine pre- 58C25 evolute and pre-curve and establish the relationship between singularities of these objects and geometric invariants of plane curves. Keywords: © 2011 Elsevier B.V. All rights reserved. Equi-centro-affine differential geometry Bifurcation set Discriminant set 1. Introduction In [5], the authors studied differential invariants of generic convex plane curves under the action of the equi-affine group on the plane R2, as an application of the singularity theory applied to equi-affine invariant functions. The equi-affine group is the special linear group SL(2, R) combined with the group of translations of the plane. Differential geometry in which the underlying transformations are the equi-affine group is called equi-affine differential geometry. In [5] an appropriate equi- affine distance function and an equi-affine height function are defined for convex plane curves. Using these, the classical notions of equi-affine vertex and inflexion, and the singularities of the classical equi-affine evolute and the equi-affine normal curve can be related to the singularities of families of functions.
    [Show full text]
  • On the Affine Heat Equation for Non-Convex Curves 1
    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Number 3, July 1998, Pages 601{634 S 0894-0347(98)00262-8 ON THE AFFINE HEAT EQUATION FOR NON-CONVEX CURVES SIGURD ANGENENT, GUILLERMO SAPIRO, AND ALLEN TANNENBAUM 1. Introduction In the past several years, there has been much research devoted to the study of evolutions of plane curves where the velocity of the evolving curve is given by the Euclidean curvature vector. This evolution appears in a number of different pure and applied areas such as differential geometry, crystal growth, and computer vision. See for example [4, 5, 6, 15, 16, 17, 19, 20, 35] and the references therein. As is well known, this Euclidean curvature evolution is a “Euclidean curve short- ening” process, precisely because the flow lines in the space of curves are tangent to the gradient of the length functional. Therefore, the curve perimeter is shrinking as fast as possible [17]. The behavior of an embedded curve evolving according to this flow has been well-studied. Gage and Hamilton prove that a convex embedded smooth initial curve converges to a round point under this evolution [13, 14, 15]. Grayson [16] has shown that a non-convex embedded curve converges to a convex one, and from there to a round point according to the Gage and Hamilton result. Since this evolution is based on Euclidean invariant concepts (Euclidean curvature vector), the solution is invariant only under rigid plane motions (i.e., the group of proper Euclidean motions in R2 generated by rotations and translations). This equation has also been called the geometric heat equation.
    [Show full text]
  • Basics of the Differential Geometry of Curves
    Chapter 19 Basics of the Differential Geometry of Curves 19.1 Introduction: Parametrized Curves In this chapter we consider parametric curves, and we introduce two important in- variants, curvature and torsion (in the case of a 3D curve). Properties of curves can be classified into local properties and global properties. Local properties are the properties that hold in a small neighborhood of a point on acurve.Curvatureisalocalproperty.Localpropertiescanbe studied more con- veniently by assuming that the curve is parametrized locally. Thus, it is important and useful to study parametrized curves. In order to study theglobalpropertiesofa curve, such as the number of points where the curvature is extremal, the number of times that a curve wraps around a point, or convexity properties, topological tools are needed. A proper study of global properties of curves really requires the intro- duction of the notion of a manifold, a concept beyond the scopeofthisbook.In this chapter we study only local properties of parametrized curves. Readers inter- ested in learning about curves as manifolds and about global properties of curves are referred to do Carmo [7] and Berger and Gostiaux [2]. Kreyszig [15] is also an excellent source, which does a great job at tracing the originofconcepts.Itturnsout that it is easier to study the notions of curvature and torsionifacurveisparametrized by arc length, and thus we will discuss briefly the notion of arclength. Let E be some normed affine space of finite dimension, for the sake of simplicity the Euclidean space E2 or E3.RecallthattheEuclideanspaceEm is obtained from the affine space Am by defining on the vector space Rm the standard inner product (x1,...,xm) (y1,...,ym)=x1y1 + + xmym.
    [Show full text]
  • Moving Frames, the Theory of Continuous Groups and Generalized Spaces
    “La méthode du repère mobile, la théorie des groupes continues et les espaces généralisés,” Exposés de géométrie, t. V, Hermann, Paris, 1935. THE METHOD OF MOVING FRAMES, THE THEORY OF CONTINUOUS GROUPS AND GENERALIZED SPACES By Élie CARTAN Translated by D. H. Delphenich _________ TABLE OF CONTENTS _________ Page INTRODUCTION…………………………………………………………………. 1 I. − The method of moving frames…………………………………………… 2 II.– Application of the method of moving frames to the study of minimal curves…………………………………………………………………….. 3 III. – The theory of plane curves in affine geometry…………………………... 10 IV. − The method of reduced equations………………………………………... 14 V. − The notion of frame in a geometry with a given group………………….. 18 VI. − The Darboux-Maurer-Cartan structure equations……………………….. 24 VII. − General method of moving frames……………………………………..... 29 VIII. – The method of moving frames, based upon the structure equations…….. 34 IX. – The structure equations, imagined as an organizing principle for space… 38 X. – Generalized spaces………………………………………………………. 42 INTRODUCTION The following pages constitute the development of five conference talks that were made in Moscow from 16 to 20 June 1930 at the invitation of the Moscow Mathematical Institute. They were translated and published in Russian in 1933. It does not seem pointless to me to insert them in the collection of Exposés de géométrie , in the hopes of submitting them for the approval of a larger number of geometers. Some theorems that required technical knowledge from the theory of partial differential equations have been stated without proof. ELIE CARTAN _____________ 2 I 1. – All of the geometry that has been known in the study of curves and surface since G. Darboux has been derived by the use of a moving trihedron that is attached to the different points of the curve or surface according to some intrinsic law ( 1).
    [Show full text]
  • Classification of Solitons for the Affine Curvature Flow
    COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 7, Number 4, 731-753, 1999 Classification of Solitons for the Affine Curvature Flow LEVI LOPES DE LIMA AND JOSE FABIO MONTENEGRO1 The curvature flow in planar affine geometry is introduced and the classification of solitons (homothetic solutions) for this flow is carried out via the method of adjoint orbits. This technique is used to integrate, up to a single quadrature, the soliton equations in terms of elliptic functions. In particular, it is shown that ellipses are the only embedded solitons. 1. Introduction. In recent years there has been much interest in investigating the asymptotic behavior of curves in the euclidean plane R2 evolving according to a flow of the type Xt = W. Here, X = X(t, u) : [0, T) x I —► R2 is a one-parameter family of closed curves and W is a vector field defined in some neighborhood of Xo = X(0,.). Usually, W is the gradient vector field of some geometric functional £ defined in the space of curves X : / —> R2. Flows studied so far include: • £(X) = length of (X). This is the curve shortening flow ([AL], [GH], [G]). Some generalizations of this flow have also deserved attention ([GL],[A1]). • £(X) = total squared curvature of X. This is the curve straightening flow ([LS],[L]). The general strategy to understand these flows can be organized in the following steps: (i) classification of solitons (homothetic solutions); (ii) lin- ear stability analysis around the solitons; (iii) local and global existence of 1Both authors have been partially supported by FUNCAP/CE. 731 732 Levi Lopes de Lima and Jose Fabio Montenegro solutions (possibly with some restrictive condition on the initial data); (iv) asymptotic behavior of solutions and corresponding estimates on the basins of attraction of solitons.
    [Show full text]
  • Affine Arc Length Polylines and Curvature Continuous Uniform B
    *Manuscript Click here to view linked References Affine arc length polylines and curvature continuous uniform B-splines Florian Kaferb¨ ock¨ Institut f¨urDiskrete Mathematik und Geometrie, Technische Universit¨atWien Wiedner Hauptstr. 8–10/104, A-1040 Wien, Austria Abstract We study the recently introduced notion of polylines that form a discrete version of planar curves in affine arc length parametrization, showing that they match the control polylines of curvature continuous uniform quadratic B-splines (with analogous results in Rn). It is demonstrated how inflection-free planar curves may be approximated by such affine arc length polylines in a way that the polyline is close to an affinely equidistant discretization of the curve and allows good approximations of the smooth affine curvature. Keywords: Discrete affine differential geometry, quadratic B-spline, geometric continuity, curve approximation 1. Introduction and Overview Affine geometry is the study of geometric objects and properties that are invariant under (equi-)affine transformations. Many of the concepts from Euclidean planar curve geometry can be reproduced in an analogous way in planar affine geometry; thus we have affine arc length, affine curve normal, affine curvature, affine evolutes, affine osculating conics (analogous to the Euclidean osculating circle). Inflection-free planar curves can be parametrized by the affine arc length. This parametrization has simple expressions for the affine normal, curvature etc., as opposed to the rather unwieldy formulae in a general parametrization. There is a geometrically simple discrete analogon to planar curves in affine arc length para- metrization: Polylines where the area spanned by three consecutive vertices is constant, that is det(xi xi 1; xi+1 xi) = A, or equivalently, where xi+1 xi xi+2 xi 1.
    [Show full text]
  • Solitons of the Midpoint Mapping and Affine Curvature
    J. Geom. (2021) 112:7 c 2021 The Author(s) 0047-2468/21/010001-16 published online January 25, 2021 Journal of Geometry https://doi.org/10.1007/s00022-020-00567-y Solitons of the midpoint mapping and affine curvature Christine Rademacher and Hans-Bert Rademacher n Abstract. For a polygon x =(xj )j∈Z in R we consider the midpoints polygon (M(x))j =(xj + xj+1) /2. We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the poly- gon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on Rn. These smooth curves are also characterized as solutions of the differential equationc ˙(t)=Bc(t)+d for a matrix B and a vector d. For n = 2 these curves are curves of constant generalized-affine cur- vature kga = kga(B) depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola. Mathematics Subject Classification. 51M04 (15A16 53A15). Keywords. Discrete curve shortening, polygon, affine mappings, soliton, midpoints polygon, linear system of ordinary differential equations. 1. Introduction n We consider an infinite polygon (xj)j∈Z given by its vertices xj ∈ R in an n-dimensional real vector space Rn resp. an n-dimensional affine space An n modelled after R . For a parameter α ∈ (0, 1) we introduce the polygon Mα(x) whose vertices are given by M x − α x αx .
    [Show full text]
  • General-Affine Invariants of Plane Curves and Space Curves
    General-affine invariants of plane curves and space curves Shimpei Kobayashi 1 Takeshi Sasaki 2 ∗ and [email protected] [email protected] 1Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan 2Department of Mathematics, Kobe University, Kobe, 657-8501, Japan September 16, 2019 Abstract We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups GA(2) = GL(2, R)⋉R2 and GA(3) = GL(3, R) ⋉ R3, respectively. We define general-affine length parameter and curva- tures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and de- rive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective treatment of curves. 1 Introduction Let An be the affine n-space with coordinates x = (x1,...,xn). It is called a unimod- ular affine space if it is equipped with a parallel volume element, namely a determinant arXiv:1902.10926v2 [math.DG] 13 Sep 2019 function. The unimodular affine group SA(n) = SL(n, R) ⋉ Rn acts as x =(xi) gx + a = gixj + ai , g =(gi) SL(n, R), a =(ai) Rn, −→ j j ∈ ∈ j X which preserves the volume element. Study of geometric properties of submanifolds in An invariant under this group is called equiaffine differential geometry, while the study of properties invariant under the general affine group GA(n) = GL(n, R) ⋉ Rn is called 2010 Mathematics subject classification.
    [Show full text]