Riemannian geometries on spaces of plane curves (original) (raw)

Reparameterization Invariant Metric on the Space of Curves

Lecture Notes in Computer Science, 2015

This paper focuses on the study of open curves in a manifold M , and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions M = Imm([0, 1], M) by pullback of a metric on the tangent bundle TM derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on TM induces a first-order Sobolev metric on M with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the SRV representations of the curves. The geodesic equations for this metric are given, as well as an idea of how to compute the exponential map for observed trajectories in applications. This provides a generalized theoretical SRV framework for curves lying in a general manifold M .

Properties of Sobolev-type metrics in the space of curves

2008

We define a manifold M where objects c ∈ M are curves, which we parameterize as c : S 1 → R n (n 2, S 1 is the circle). We study geometries on the manifold of curves, provided by Sobolevtype Riemannian metrics H j. These metrics have been shown to regularize gradient flows used in computer vision applications (see [13, 14, 16] and references therein). We provide some basic results on H j metrics; and, for the cases j = 1, 2, we characterize the completion of the space of smooth curves. We call these completions "H 1 and H 2 Sobolev-type Riemannian manifolds of curves". This result is fundamental since it is a first step in proving the existence of geodesics with respect to these metrics. As a byproduct, we prove that the Fréchet distance of curves (see [7]) coincides with the distance induced by the "Finsler L ∞ metric" defined in §2.2 of [18].

The curvature: a variational approach

The curvature discussed in this paper is a far reaching generalisation of the Riemannian sectional curvature. We give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optimal control problems, including Riemannian, sub-Riemannian, Finsler and sub-Finsler spaces. Special attention is paid to the sub-Riemannian (or Carnot-Carathéodory) metric spaces. Our construction of curvature is direct and naive, and similar to the original approach of Riemann. In particular, we extract geometric invariants from the asymptotics of the cost of optimal control problems. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces. Contents Chapter 1. Introduction v 1.1. Structure of the paper viii 1.2. Statements of the main theorems viii 1.3. The Heisenberg group x Part 1. Statements of the results Chapter 2. General setting 2.1. Affine control systems 2.2. End-point map 2.3. Lagrange multipliers rule 2.4. Pontryagin Maximum Principle 2.5. Regularity of the value function Chapter 3. Flag and growth vector of an admissible curve 3.1. Growth vector of an admissible curve 3.2. Linearised control system and growth vector 3.3. State-feedback invariance of the flag of an admissible curve 3.4. An alternative definition Chapter 4. Geodesic cost and its asymptotics 4.1. Motivation: a Riemannian interlude 4.2. Geodesic cost 4.3. Hamiltonian inner product 4.4. Asymptotics of the geodesic cost function and curvature 4.5. Examples Chapter 5. Sub-Riemannian geometry 5.1. Basic definitions 5.2. Existence of ample geodesics 5.3. Reparametrization and homogeneity of the curvature operator 5.4. Asymptotics of the sub-Laplacian of the geodesic cost 5.5. Equiregular distributions 5.6. Geodesic dimension and sub-Riemannian homotheties 5.7. Heisenberg group 5.8. On the "meaning" of constant curvature Part 2. Technical tools and proofs Chapter 6. Jacobi curves 6.1. Curves in the Lagrange Grassmannian 6.2. The Jacobi curve and the second differential of the geodesic cost 6.3. The Jacobi curve and the Hamiltonian inner product 6.4. Proof of Theorem A 6.5. Proof of Theorem D Chapter 7. Asymptotics of the Jacobi curve: equiregular case 7.1. The canonical frame 7.2. Main result iii 7.3. Proof of Theorem 7.4 7.4. Proof of Theorem B 7.5. A worked out example: 3D contact sub-Riemannian structures Chapter 8. Sub-Laplacian and Jacobi curves 8.1. Coordinate lift of a local frame 8.2. Sub-Laplacian of the geodesic cost 8.3. Proof of Theorem C

Reparameterization invariant distance on the space of curves in the hyperbolic plane

AIP Conference Proceedings, 2015

This paper focuses on the study of time-varying paths in the two-dimensional hyperbolic space, and its aim is to define a reparameterization invariant distance on the space of such paths. We adapt the geodesical distance on the space of parameterized plane curves given by Bauer et al. in [1] to the space Imm([0, 1], H) of parameterized curves in the hyperbolic plane. We present a definition which enables to evaluate the difference between two curves, and show that it satisfies the three properties of a metric. Unlike the distance of Bauer et al., the distance obtained takes into account the positions of the curves, and not only their shapes and parameterizations, by including the distance between their origins.

Curvature: A Variational Approach

Memoirs of the American Mathematical Society

METRICS OF CURVED SURFACES AND SPACES

The word metric is derived from the Greek word metria meaning measurement. Historically, its origin lies in the measurement of distances on the surface of the Earth so that inundations in the river Nile could be curbed out. Before introducing metrics of a curved surface and spaces, we examine the formula for linear distance between two points in a plane, which is essentially based on the celebrated Pythagoras theorem. Given two infinitesimal points P, Q with rectangular Cartesian coordinates (x, y) and (x + dx, y + dy) in a plane the distance PQ ≡ ds between them is measured by (ds) 2 = (PC) 2 + (CQ) 2 = (OB – OA) 2 + (BQ – BC) 2 = (dx) 2 + (dy) 2. (1.1) The right member of above equation is called the metric of the plane E 2. Above formula can be also generalized to a three-dimensional Euclidean space E 3 as well as to a Euclidean space E n of arbitrary dimension n ε N. In the next Section, we first obtain a formula for the metric of a curved surface V 2 immersed in the space E 3 and then deduce (1.1) from the same as a special case. § 2. Metric of a V 2 Let V 2 be a curved surface immersed in the Euclidean space E 3. The position vector r of a point P ε V 2 with respect to the origin O in E 3 , is thus, expressible as a function of two independent parameters, say u 1 and u 2 : r = r (u 1 , u 2). (2.1) The sets of points of V 2 for which one of the parameters remains constant describe the parametric curves: u 1-curve (along which u 2 = const.), u 2-curve (along which u 1 = const.). Thus, the parametric curves having equations u 2 = const. and u 1 = const. (2.2) respectively form a curvilinear coordinate system on V 2. Accordingly, (2.1) determines the position vectors r = r (u 1) and r = r (u 2) of the points on respective

An Introduction to Riemannian Geometry - Lecture Notes in Mathematics

These lecture notes grew out of an M.Sc. course on differential geometry which I gave at the University of Leeds 1992. Their main purpose is to introduce the beautiful theory of Riemannian Geometry a still very active area of mathematical research. This is a subject with no lack of interesting examples. They are indeed the key to a good understanding of it and will therefore play a major role throughout this work. Of special interest are the classical Lie groups allowing concrete calculations of many of the abstract notions on the menu. The study of Riemannian geometry is rather meaningless without some basic knowledge on Gaussian geometry that i.e. the geometry of curves and surfaces in 3-dimensional space. For this I recommend the excellent textbook: M. P. do Carmo, Differential geometry of curves and surfaces, Prentice Hall (1976). These lecture notes are written for students with a good understanding of linear algebra, real analysis of several variables, the classical theory of ordinary differential equations and some topology. The most important results stated in the text are also proven there. Others are left to the reader as exercises, which follow at the end of each chapter. This format is aimed at students willing to put hard work into the course. For further reading I recommend the interesting textbook: M. P. do Carmo, Riemannian Geometry, Birkhäuser (1992). I am grateful to my many enthusiastic students who throughout the years have contributed to the text by finding numerous typing errors and giving many useful comments on the presentation.

On the Generalized Curvature

AL-Rafidain Journal of Computer Sciences and Mathematics

By using methods of nonstandard analysis given by Robinson, A., and axiomatized by Nelson, E., we try in this paper to establish the generalized curvature of a plane curve () t  at regular points and at points infinitely close to a singular point. It is known that the radius of curvature of a plane curve () t  is the limit of the radius of a circle circumscribed to a triangle ABC, where B and C are points of  infinitely close to A. Our goal is to give a nonstandard proof of this fact. More precisely, if A is a standard point of a standard curve  and B, C are points of  defined by