Riemannian Geometry: Some Examples, Including Map Projections
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Map Projections--A Working Manual
This is a reproduction of a library book that was digitized by Google as part of an ongoing effort to preserve the information in books and make it universally accessible. https://books.google.com 7 I- , t 7 < ?1 > I Map Projections — A Working Manual By JOHN P. SNYDER U.S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 1395 _ i UNITED STATES GOVERNMENT PRINTING OFFIGE, WASHINGTON: 1987 no U.S. DEPARTMENT OF THE INTERIOR BRUCE BABBITT, Secretary U.S. GEOLOGICAL SURVEY Gordon P. Eaton, Director First printing 1987 Second printing 1989 Third printing 1994 Library of Congress Cataloging in Publication Data Snyder, John Parr, 1926— Map projections — a working manual. (U.S. Geological Survey professional paper ; 1395) Bibliography: p. Supt. of Docs. No.: I 19.16:1395 1. Map-projection — Handbooks, manuals, etc. I. Title. II. Series: Geological Survey professional paper : 1395. GA110.S577 1987 526.8 87-600250 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, DC 20402 PREFACE This publication is a major revision of USGS Bulletin 1532, which is titled Map Projections Used by the U.S. Geological Survey. Although several portions are essentially unchanged except for corrections and clarification, there is consider able revision in the early general discussion, and the scope of the book, originally limited to map projections used by the U.S. Geological Survey, now extends to include several other popular or useful projections. These and dozens of other projections are described with less detail in the forthcoming USGS publication An Album of Map Projections. As before, this study of map projections is intended to be useful to both the reader interested in the philosophy or history of the projections and the reader desiring the mathematics. -
Geometry of Curves
Appendix A Geometry of Curves “Arc, amplitude, and curvature sustain a similar relation to each other as time, motion and velocity, or as volume, mass and density.” Carl Friedrich Gauss The rest of this lecture notes is about geometry of curves and surfaces in R2 and R3. It will not be covered during lectures in MATH 4033 and is not essential to the course. However, it is recommended for readers who want to acquire workable knowledge on Differential Geometry. A.1. Curvature and Torsion A.1.1. Regular Curves. A curve in the Euclidean space Rn is regarded as a function r(t) from an interval I to Rn. The interval I can be finite, infinite, open, closed n n or half-open. Denote the coordinates of R by (x1, x2, ... , xn), then a curve r(t) in R can be written in coordinate form as: r(t) = (x1(t), x2(t),..., xn(t)). One easy way to make sense of a curve is to regard it as the trajectory of a particle. At any time t, the functions x1(t), x2(t), ... , xn(t) give the coordinates of the particle in n R . Assuming all xi(t), where 1 ≤ i ≤ n, are at least twice differentiable, then the first derivative r0(t) represents the velocity of the particle, its magnitude jr0(t)j is the speed of the particle, and the second derivative r00(t) represents the acceleration of the particle. As a course on Differential Manifolds/Geometry, we will mostly study those curves which are infinitely differentiable (i.e. -
21. Orthonormal Bases
21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else. -
Abstract Tensor Systems and Diagrammatic Representations
Abstract tensor systems and diagrammatic representations J¯anisLazovskis September 28, 2012 Abstract The diagrammatic tensor calculus used by Roger Penrose (most notably in [7]) is introduced without a solid mathematical grounding. We will attempt to derive the tools of such a system, but in a broader setting. We show that Penrose's work comes from the diagrammisation of the symmetric algebra. Lie algebra representations and their extensions to knot theory are also discussed. Contents 1 Abstract tensors and derived structures 2 1.1 Abstract tensor notation . 2 1.2 Some basic operations . 3 1.3 Tensor diagrams . 3 2 A diagrammised abstract tensor system 6 2.1 Generation . 6 2.2 Tensor concepts . 9 3 Representations of algebras 11 3.1 The symmetric algebra . 12 3.2 Lie algebras . 13 3.3 The tensor algebra T(g)....................................... 16 3.4 The symmetric Lie algebra S(g)................................... 17 3.5 The universal enveloping algebra U(g) ............................... 18 3.6 The metrized Lie algebra . 20 3.6.1 Diagrammisation with a bilinear form . 20 3.6.2 Diagrammisation with a symmetric bilinear form . 24 3.6.3 Diagrammisation with a symmetric bilinear form and an orthonormal basis . 24 3.6.4 Diagrammisation under ad-invariance . 29 3.7 The universal enveloping algebra U(g) for a metrized Lie algebra g . 30 4 Ensuing connections 32 A Appendix 35 Note: This work relies heavily upon the text of Chapter 12 of a draft of \An Introduction to Quantum and Vassiliev Invariants of Knots," by David M.R. Jackson and Iain Moffatt, a yet-unpublished book at the time of writing. -
Multilinear Algebra and Applications July 15, 2014
Multilinear Algebra and Applications July 15, 2014. Contents Chapter 1. Introduction 1 Chapter 2. Review of Linear Algebra 5 2.1. Vector Spaces and Subspaces 5 2.2. Bases 7 2.3. The Einstein convention 10 2.3.1. Change of bases, revisited 12 2.3.2. The Kronecker delta symbol 13 2.4. Linear Transformations 14 2.4.1. Similar matrices 18 2.5. Eigenbases 19 Chapter 3. Multilinear Forms 23 3.1. Linear Forms 23 3.1.1. Definition, Examples, Dual and Dual Basis 23 3.1.2. Transformation of Linear Forms under a Change of Basis 26 3.2. Bilinear Forms 30 3.2.1. Definition, Examples and Basis 30 3.2.2. Tensor product of two linear forms on V 32 3.2.3. Transformation of Bilinear Forms under a Change of Basis 33 3.3. Multilinear forms 34 3.4. Examples 35 3.4.1. A Bilinear Form 35 3.4.2. A Trilinear Form 36 3.5. Basic Operation on Multilinear Forms 37 Chapter 4. Inner Products 39 4.1. Definitions and First Properties 39 4.1.1. Correspondence Between Inner Products and Symmetric Positive Definite Matrices 40 4.1.1.1. From Inner Products to Symmetric Positive Definite Matrices 42 4.1.1.2. From Symmetric Positive Definite Matrices to Inner Products 42 4.1.2. Orthonormal Basis 42 4.2. Reciprocal Basis 46 4.2.1. Properties of Reciprocal Bases 48 4.2.2. Change of basis from a basis to its reciprocal basis g 50 B B III IV CONTENTS 4.2.3. -
Arxiv:1811.01571V2 [Cs.CV] 24 Jan 2019 Many Traditional Cnns on 3D Data Simply Extend the 2D Convolutional Op- Erations to 3D, for Example, the Work of Wu Et Al
SPNet: Deep 3D Object Classification and Retrieval using Stereographic Projection Mohsen Yavartanoo1[0000−0002−0109−1202], Eu Young Kim1[0000−0003−0528−6557], and Kyoung Mu Lee1[0000−0001−7210−1036] Department of ECE, ASRI, Seoul National University, Seoul, Korea https://cv.snu.ac.kr/ fmyavartanoo, shreka116, [email protected] Abstract. We propose an efficient Stereographic Projection Neural Net- work (SPNet) for learning representations of 3D objects. We first trans- form a 3D input volume into a 2D planar image using stereographic projection. We then present a shallow 2D convolutional neural network (CNN) to estimate the object category followed by view ensemble, which combines the responses from multiple views of the object to further en- hance the predictions. Specifically, the proposed approach consists of four stages: (1) Stereographic projection of a 3D object, (2) view-specific fea- ture learning, (3) view selection and (4) view ensemble. The proposed ap- proach performs comparably to the state-of-the-art methods while having substantially lower GPU memory as well as network parameters. Despite its lightness, the experiments on 3D object classification and shape re- trievals demonstrate the high performance of the proposed method. Keywords: 3D object classification · 3D object retrieval · Stereographic Projection · Convolutional Neural Network · View Ensemble · View Se- lection. 1 Introduction In recent years, success of deep learning methods, in particular, convolutional neural network (CNN), has urged rapid development in various computer vision applications such as image classification, object detection, and super-resolution. Along with the drastic advances in 2D computer vision, understanding 3D shapes and environment have also attracted great attention. -
Geometric-Algebra Adaptive Filters Wilder B
1 Geometric-Algebra Adaptive Filters Wilder B. Lopes∗, Member, IEEE, Cassio G. Lopesy, Senior Member, IEEE Abstract—This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are Faces generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well- Edges suited for the description of geometric transformations. Also, (directed lines) differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows Fig. 1. A polyhedron (3-dimensional polytope) can be completely described for applying the same derivation techniques regardless of the by the geometric multiplication of its edges (oriented lines, vectors), which type (subalgebra) of the data, i.e., real, complex numbers, generate the faces and hypersurfaces (in the case of a general n-dimensional quaternions, etc. Relying on those characteristics (among others), polytope). a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares perform calculus with hypercomplex quantities, i.e., elements (LMS) adaptive filter variants: real-entries LMS, complex LMS, that generalize complex numbers for higher dimensions [2]– and quaternions LMS. Mean-square analysis and simulations in [10]. a system identification scenario are provided, showing very good agreement for different levels of measurement noise. GA-based AFs were first introduced in [11], [12], where they were successfully employed to estimate the geometric Index Terms—Adaptive filtering, geometric algebra, quater- transformation (rotation and translation) that aligns a pair of nions. -
Matrices and Tensors
APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM.. -
Uon Digital Repository Home
University of Nairobi School of Engineering Building a QGIS Helper Application to Overcome the Challenges of Cassini to UTM Coordinate System Conversions in Kenya BY Ajanga Dissent Ingati F56/81982/2015 A Project submitted in partial fulfillment for the Degree of Master of Science in Geographical Information Systems, in the Department of Geospatial & Space Technology of the University of Nairobi October 2017 I Declaration I, Ajanga Dissent Ingati, hereby declare that this project is my original work. To the best of my knowledge, the work presented here has not been presented for a degree in any other Institution of Higher Learning. ……………………………………. ……………………… .……..……… Name of student Signature Date This project has been submitted for examination with my approval as university supervisor. ……………………………………. ……………………… .……..……… Name of Supervisor Signature Date I Dedication I dedicate this work to my family for their support throughout the entire process of pursuing this degree and to the completion of the project. Your encouragement has been instrumental in ensuring the success of this project; may God truly bless you. II Acknowledgement I would like to show my heartfelt appreciation to all the people who saw me through the process of preparing this report. These thanks go to all those who provided technical support, talked through the issues I studied, read, wrote, gave comments, and also those who assisted in editing and proof reading of this project report before submission. I would like to thank Steve Firsake for his technical support through Python Programing. Special thanks also go to Dr.-Ing. Musyoka for his technical advice and support for the entire project. -
Riemannian Submanifolds: a Survey
RIEMANNIAN SUBMANIFOLDS: A SURVEY BANG-YEN CHEN Contents Chapter 1. Introduction .............................. ...................6 Chapter 2. Nash’s embedding theorem and some related results .........9 2.1. Cartan-Janet’s theorem .......................... ...............10 2.2. Nash’s embedding theorem ......................... .............11 2.3. Isometric immersions with the smallest possible codimension . 8 2.4. Isometric immersions with prescribed Gaussian or Gauss-Kronecker curvature .......................................... ..................12 2.5. Isometric immersions with prescribed mean curvature. ...........13 Chapter 3. Fundamental theorems, basic notions and results ...........14 3.1. Fundamental equations ........................... ..............14 3.2. Fundamental theorems ............................ ..............15 3.3. Basic notions ................................... ................16 3.4. A general inequality ............................. ...............17 3.5. Product immersions .............................. .............. 19 3.6. A relationship between k-Ricci tensor and shape operator . 20 3.7. Completeness of curvature surfaces . ..............22 Chapter 4. Rigidity and reduction theorems . ..............24 4.1. Rigidity ....................................... .................24 4.2. A reduction theorem .............................. ..............25 Chapter 5. Minimal submanifolds ....................... ...............26 arXiv:1307.1875v1 [math.DG] 7 Jul 2013 5.1. First and second variational formulas -
Some Examples of Critical Points for the Total Mean Curvature Functional
Proceedings of the Edinburgh Mathematical Society (2000) 43, 587-603 © SOME EXAMPLES OF CRITICAL POINTS FOR THE TOTAL MEAN CURVATURE FUNCTIONAL JOSU ARROYO1, MANUEL BARROS2 AND OSCAR J. GARAY1 1 Departamento de Matemdticas, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea, Aptdo 644 > 48080 Bilbao, Spain ([email protected]; [email protected]) 2 Departamento de Geometria y Topologia, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain ([email protected]) (Received 31 March 1999) Abstract We study the following problem: existence and classification of closed curves which are critical points for the total curvature functional, defined on spaces of curves in a Riemannian manifold. This problem is completely solved in a real space form. Next, we give examples of critical points for this functional in a class of metrics with constant scalar curvature on the three sphere. Also, we obtain a rational one-parameter family of closed helices which are critical points for that functional in CP2(4) when it is endowed with its usual Kaehlerian structure. Finally, we use the principle of symmetric criticality to get equivariant submanifolds, constructed on the above curves, which are critical points for the total mean curvature functional. Keywords: total curvature; critical points; total tension AMS 1991 Mathematics subject classification: Primary 53C40; 53A05 1. Introduction The tension field of a submanifold is the Euler-Lagrange operator associated with the energy of the submanifold [9] and it is precisely the mean curvature vector field. Thus, the amount of tension that a submanifold receives at one point is measured by the mean curvature function at that point, a. -
Arxiv:1709.02078V1 [Math.DG]
DENSITY OF A MINIMAL SUBMANIFOLD AND TOTAL CURVATURE OF ITS BOUNDARY Jaigyoung Choe∗ and Robert Gulliver Abstract Given a piecewise smooth submanifold Γn−1 ⊂ Rm and p ∈ Rm, we define the vision angle Πp(Γ) to be the (n − 1)-dimensional volume of the radial projection of Γ to the unit sphere centered at p. If p is a point on a stationary n-rectifiable set Σ ⊂ Rm with boundary Γ, then we show the density of Σ at p is ≤ the density at its vertex p of the n−1 cone over Γ. It follows that if Πp(Γ) is less than twice the volume of S , for all p ∈ Γ, then Σ is an embedded submanifold. As a consequence, we prove that given two n-planes n n Rm n R1 , R2 in and two compact convex hypersurfaces Γi of Ri ,i =1, 2, a nonflat minimal submanifold spanned by Γ := Γ1 ∪ Γ2 is embedded. 1 Introduction Fenchel [F1] showed that the total curvature of a closed space curve γ ⊂ Rm is at least 2π, and it equals 2π if and only if γ is a plane convex curve. F´ary [Fa] and Milnor [M] independently proved that a simple knotted regular curve has total curvature larger than 4π. These two results indicate that a Jordan curve which is curved at most double the minimum is isotopically simple. But in fact minimal surfaces spanning such Jordan curves must be simple as well. Indeed, Nitsche [N] showed that an analytic Jordan curve in R3 with total curvature at most 4π bounds exactly one minimal disk.