A Disorienting Look at Euler's Theorem on the Axis of a Rotation
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Mostly Surfaces Richard Evan Schwartz
Mostly Surfaces Richard Evan Schwartz Department of Mathematics, Brown University Current address: 151 Thayer St. Providence, RI 02912 E-mail address: [email protected] 2010 Mathematics Subject Classification. Primary Key words and phrases. surfaces, geometry, topology, complex analysis supported by N.S.F. grant DMS-0604426. Contents Preface xiii Chapter 1. Book Overview 1 1.1. Behold, the Torus! 1 § 1.2. Gluing Polygons 3 § 1.3.DrawingonaSurface 5 § 1.4. Covering Spaces 8 § 1.5. HyperbolicGeometryandtheOctagon 9 § 1.6. Complex Analysis and Riemann Surfaces 11 § 1.7. ConeSurfacesandTranslationSurfaces 13 § 1.8. TheModularGroupandtheVeechGroup 14 § 1.9. Moduli Space 16 § 1.10. Dessert 17 § Part 1. Surfaces and Topology Chapter2. DefinitionofaSurface 21 2.1. A Word about Sets 21 § 2.2. Metric Spaces 22 § 2.3. OpenandClosedSets 23 § 2.4. Continuous Maps 24 § v vi Contents 2.5. Homeomorphisms 25 § 2.6. Compactness 26 § 2.7. Surfaces 26 § 2.8. Manifolds 27 § Chapter3. TheGluingConstruction 31 3.1. GluingSpacesTogether 31 § 3.2. TheGluingConstructioninAction 34 § 3.3. The Classification of Surfaces 36 § 3.4. TheEulerCharacteristic 38 § Chapter 4. The Fundamental Group 43 4.1. A Primer on Groups 43 § 4.2. Homotopy Equivalence 45 § 4.3. The Fundamental Group 46 § 4.4. Changing the Basepoint 48 § 4.5. Functoriality 49 § 4.6. Some First Steps 51 § Chapter 5. Examples of Fundamental Groups 53 5.1.TheWindingNumber 53 § 5.2. The Circle 56 § 5.3. The Fundamental Theorem of Algebra 57 § 5.4. The Torus 58 § 5.5. The 2-Sphere 58 § 5.6. TheProjectivePlane 59 § 5.7. A Lens Space 59 § 5.8. -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
(II) I Main Topics a Rotations Using a Stereonet B
GG303 Lecture 11 9/4/01 1 ROTATIONS (II) I Main Topics A Rotations using a stereonet B Comments on rotations using stereonets and matrices C Uses of rotation in geology (and engineering) II II Rotations using a stereonet A Best uses 1 Rotation axis is vertical 2 Rotation axis is horizontal (e.g., to restore tilted beds) B Construction technique 1 Find orientation of rotation axis 2 Find angle of rotation and rotate pole to plane (or a linear feature) along a small circle perpendicular to the rotation axis. a For a horizontal rotation axis the small circle is vertical b For a vertical rotation axis the small circle is horizontal 3 WARNING: DON'T rotate a plane by rotating its dip direction vector; this doesn't work (see handout) IIIComments on rotations using stereonets and matrices Advantages Disadvantages Stereonets Good for visualization Relatively slow Can bring into field Need good stereonets, paper Matrices Speed and flexibility Computer really required to Good for multiple rotations cut down on errors A General comments about stereonets vs. rotation matrices 1 With stereonets, an object is usually considered to be rotated and the coordinate axes are held fixed. 2 With rotation matrices, an object is usually considered to be held fixed and the coordinate axes are rotated. B To return tilted bedding to horizontal choose a rotation axis that coincides with the direction of strike. The angle of rotation is the negative of the dip of the bedding (right-hand rule! ) if the bedding is to be rotated back to horizontal and positive if the axes are to be rotated to the plane of the tilted bedding. -
Euler Quaternions
Four different ways to represent rotation my head is spinning... The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why det( R ) = ± 1 ? − = − Rotations preserve distance: Rp1 Rp2 p1 p2 Rotations preserve orientation: ( ) × ( ) = ( × ) Rp1 Rp2 R p1 p2 The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why it’s a group: • Closed under multiplication: if ∈ ( ) then ∈ ( ) R1, R2 SO 3 R1R2 SO 3 • Has an identity: ∃ ∈ ( ) = I SO 3 s.t. IR1 R1 • Has a unique inverse… • Is associative… Why orthogonal: • vectors in matrix are orthogonal Why it’s special: det( R ) = + 1 , NOT det(R) = ±1 Right hand coordinate system Possible rotation representations You need at least three numbers to represent an arbitrary rotation in SO(3) (Euler theorem). Some three-number representations: • ZYZ Euler angles • ZYX Euler angles (roll, pitch, yaw) • Axis angle One four-number representation: • quaternions ZYZ Euler Angles φ = θ rzyz ψ φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1. Rotate φ about z axis 0 0 1 θ θ 2. Then rotate θ about y axis cos 0 sin θ = ψ Ry ( ) 0 1 0 3. Then rotate about z axis − sinθ 0 cosθ ψ − ψ cos sin 0 ψ = ψ ψ Rz ( ) sin cos 0 0 0 1 ZYZ Euler Angles φ θ ψ Remember that R z ( ) R y ( ) R z ( ) encode the desired rotation in the pre- rotation reference frame: φ = pre−rotation Rz ( ) Rpost−rotation Therefore, the sequence of rotations is concatentated as follows: (φ θ ψ ) = φ θ ψ Rzyz , , Rz ( )Ry ( )Rz ( ) φ − φ θ θ ψ − ψ cos sin 0 cos 0 sin cos sin 0 (φ θ ψ ) = φ φ ψ ψ Rzyz , , sin cos 0 0 1 0 sin cos 0 0 0 1− sinθ 0 cosθ 0 0 1 − − − cφ cθ cψ sφ sψ cφ cθ sψ sφ cψ cφ sθ (φ θ ψ ) = + − + Rzyz , , sφ cθ cψ cφ sψ sφ cθ sψ cφ cψ sφ sθ − sθ cψ sθ sψ cθ ZYX Euler Angles (roll, pitch, yaw) φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1. -
(12) Patent Application Publication (10) Pub. No.: US 2015/0260527 A1 Fink (43) Pub
US 20150260527A1 (19) United States (12) Patent Application Publication (10) Pub. No.: US 2015/0260527 A1 Fink (43) Pub. Date: Sep. 17, 2015 (54) SYSTEMAND METHOD OF DETERMINING (52) U.S. Cl. A POSITION OF A REMOTE OBJECT VA CPC ...................................... G0IC 2 1/20 (2013.01) ONE ORMORE IMAGE SENSORS (57) ABSTRACT (71) Applicant: Ian Michael Fink, Austin, TX (US) In one or more embodiments, one or more systems, methods (72) Inventor: Ian Michael Fink, Austin, TX (US) and/or processes may determine a location of a remote object (e.g., a point and/or area of interest, landmark, structure that (21) Appl. No.: 14/716,817 “looks interesting, buoy, anchored boat, etc.). For example, the location of a remote object may be determined via a first (22) Filed: May 19, 2015 bearing, at a first location, and a second bearing, at a second O O location, to the remote object. For instance, the first and Related U.S. Application Data second locations can be determined via a position device, (63) Continuation of application No. 13/837,853, filed on Such as a global positioning system device. In one or more Mar. 15, 2013, now Pat. No. 9,074,892. embodiments, the location of the remote object may be based on the first location, the second location, the first bearing, and Publication Classification the second bearing. For example, the location of the remote object may be provided to a user via a map. For instance, (51) Int. Cl. turn-by-turn direction to the location of the remote object GOIC 2L/20 (2006.01) may be provided to the user. -
Spherical Geometry
Spherical Geometry Eric Lehman february-april 2012 2 Table of content 1 Spherical biangles and spherical triangles 3 § 1. The unit sphere . .3 § 2. Biangles . .6 § 3. Triangles . .7 2 Coordinates on a sphere 13 § 1. Cartesian versus spherical coordinates . 13 § 2. Basic astronomy . 16 § 3. Geodesics . 16 3 Spherical trigonometry 23 § 1. Pythagoras’ theorem . 23 § 2. The three laws of spherical trigonometry . 24 4 Stereographic projection 29 § 1. Definition . 29 § 2. The stereographic projection preserves the angles . 34 § 3. Images of circles . 36 5 Projective geometry 49 § 1. First description of the real projective plane . 49 § 2. The real projective plane . 52 § 3. Generalisations . 59 3 4 TABLE OF CONTENT TABLE OF CONTENT 1 Introduction The aim of this course is to show different aspects of spherical geometry for itself, in relation to applications and in relation to other geometries and other parts of mathematics. The chapters will be (mostly) independant from each other. To begin, we’l work on the sphere as Euclid did in the plane looking at triangles. Many things look alike, but there are some striking differences. The second viewpoint will be the introduction of coordinates and the application to basic astronomy. The theorem of Pytha- goras has a very nice and simple shape in spherical geometry. To contemplate spherical trigonometry will give us respect for our ancestors and navigators, but we shall skip the computations! and let the GPS do them. The stereographic projection is a marvellous tool to understand the pencils of coaxial circles and many aspects of the relation between the spherical geometry, the euclidean affine plane, the complex projective line, the real projec- tive plane, the Möbius strip and even the hyperbolic plane. -
The Structuring of Pattern Repeats in the Paracas Necropolis Embroideries
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Textile Society of America Symposium Proceedings Textile Society of America 1988 Orientation And Symmetry: The Structuring Of Pattern Repeats In The Paracas Necropolis Embroideries Mary Frame University of British Columbia Follow this and additional works at: https://digitalcommons.unl.edu/tsaconf Part of the Art and Design Commons Frame, Mary, "Orientation And Symmetry: The Structuring Of Pattern Repeats In The Paracas Necropolis Embroideries" (1988). Textile Society of America Symposium Proceedings. 637. https://digitalcommons.unl.edu/tsaconf/637 This Article is brought to you for free and open access by the Textile Society of America at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Textile Society of America Symposium Proceedings by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 136 RESEARCH-IN-PROGRESS REPORT ORIENTATION AND SYMMETRY: THE STRUCTURING OF PATTERN REPEATS IN THE PARACAS NECROPOLIS EMBROIDERIES MARY FRAME RESEARCH ASSOCIATE, INSTITUTE OF ANDEAN STUDIES 4611 Marine Drive West Vancouver, B.C. V7W 2P1 The most extensive Peruvian fabric remains come from the archeological site of Paracas Necropolis on the South Coast of Peru. Preserved by the dry desert conditions, this cache of 429 mummy bundles, excavated in 1926-27, provides an unparal- leled opportunity for comparing the range and nature of varia- tions in similar fabrics which are securely related in time and space. The bundles are thought to span the time period from 500-200 B.C. The most numerous and notable fabrics are embroideries: garments that have been classified as mantles, tunics, wrap- around skirts, loincloths, turbans and ponchos. -
Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation -
The Problem of Exterior Orientation in Photogrammetry
T he Problem of Exterior Orientation in Photogrammetry*t GEORGE H. ROSENFIELD, Data Reduction A nalyst, RCA Service Co., Air Force Missile Test Center, Patrick Air Force Base, Fla. ABSTRACT: Increased interest in Analytical Photogrammetry has focused at tention upon a chaotic situation which complicates research and communication between photogrammetrists. The designation of the orientation parameters in a different manner by almost every investigator has resulted in the need for a universal non-ambiguous system which depicts the actual physical orientation of the photograph. This paper presents a discussion of the basic concepts in volved in the various systems in use, and recommends the adoption of a standard system within the basic framework of the Stockholm resolution of July 1956, concerning the sign convention to be used in Photogrammetry. 1. INTRODUCTION CHAOTIC situation exists within the science of Photogrammetry which complicates A research and communication between photogrammetrists. When the theoretician studies a technical paper, he finds the work unnecessarily laborious until he has first become familiar with the coordinate system, definitions, and rotations employed by the writer. \i\Then the skilled instrument operator sits down to a newstereoplotting instrument he finds the orienta tion procedure unnecessarily difficult until he be comes familiar with the positive direction of the translational and rotational motions. Thus it is that almost every individual has devised his own unique system of designating the exterior orienta tion of the photograph. In order to alleviate these complications, the VIII International Congress for Photogrammetry at Stockholm in July 1956, accepted a resolution on the sign convention to be used in photogrammetry. -
Oriented Projective Geometry for Computer Vision
Oriented Projective Geometry for Computer Vision St~phane Laveau Olivier Faugeras e-mall: Stephane. Laveau@sophia. inria, fr, Olivier.Faugeras@sophia. inria, fr INRIA. 2004, route des Lucioles. B.R 93. 06902 Sophia-Antipolis. FRANCE. Abstract. We present an extension of the usual projective geometric framework for computer vision which can nicely take into account an information that was previously not used, i.e. the fact that the pixels in an image correspond to points which lie in front of the camera. This framework, called the oriented projective geometry, retains all the advantages of the unoriented projective geometry, namely its simplicity for expressing the viewing geometry of a system of cameras, while extending its adequation to model realistic situations. We discuss the mathematical and practical issues raised by this new framework for a number of computer vision algorithms. We present different experiments where this new tool clearly helps. 1 Introduction Projective geometry is now established as the correct and most convenient way to de- scribe the geometry of systems of cameras and the geometry of the scene they record. The reason for this is that a pinhole camera, a very reasonable model for most cameras, is really a projective (in the sense of projective geometry) engine projecting (in the usual sense) the real world onto the retinal plane. Therefore we gain a lot in simplicity if we represent the real world as a part of a projective 3-D space and the retina as a part of a projective 2-D space. But in using such a representation, we apparently loose information: we are used to think of the applications of computer vision as requiring a Euclidean space and this notion is lacking in the projective space. -
A Guided Tour to the Plane-Based Geometric Algebra PGA
A Guided Tour to the Plane-Based Geometric Algebra PGA Leo Dorst University of Amsterdam Version 1.15{ July 6, 2020 Planes are the primitive elements for the constructions of objects and oper- ators in Euclidean geometry. Triangulated meshes are built from them, and reflections in multiple planes are a mathematically pure way to construct Euclidean motions. A geometric algebra based on planes is therefore a natural choice to unify objects and operators for Euclidean geometry. The usual claims of `com- pleteness' of the GA approach leads us to hope that it might contain, in a single framework, all representations ever designed for Euclidean geometry - including normal vectors, directions as points at infinity, Pl¨ucker coordinates for lines, quaternions as 3D rotations around the origin, and dual quaternions for rigid body motions; and even spinors. This text provides a guided tour to this algebra of planes PGA. It indeed shows how all such computationally efficient methods are incorporated and related. We will see how the PGA elements naturally group into blocks of four coordinates in an implementation, and how this more complete under- standing of the embedding suggests some handy choices to avoid extraneous computations. In the unified PGA framework, one never switches between efficient representations for subtasks, and this obviously saves any time spent on data conversions. Relative to other treatments of PGA, this text is rather light on the mathematics. Where you see careful derivations, they involve the aspects of orientation and magnitude. These features have been neglected by authors focussing on the mathematical beauty of the projective nature of the algebra. -
Correlators of Wilson Loops on Hopf Fibrations
UNIVERSITA` DEGLI STUDI DI PARMA Dottorato di Ricerca in Fisica Ciclo XXV Correlators of Wilson loops on Hopf fibrations in the AdS5=CFT4 correspondence Coordinatore: Chiar.mo Prof. PIERPAOLO LOTTICI Supervisore: Dott. LUCA GRIGUOLO Dottorando: STEFANO MORI Anno Accademico 2012/2013 ויאמר אלהיM יהי אור ויהי אור בראשׂיה 1:3 Io stimo pi`uil trovar un vero, bench´edi cosa leggiera, che 'l disputar lungamente delle massime questioni senza conseguir verit`anissuna. (Galileo Galilei, Scritti letterari) Contents 1 The correspondence and its observables 1 1.1 AdS5=CFT4 correspondence . .1 1.1.1 N = 4 SYM . .1 1.1.2 AdS space . .2 1.2 Brane construction . .4 1.3 Symmetries matching . .6 1.4 AdS/CFT dictionary . .7 1.5 Integrability . .9 1.6 Wilson loops and Minimal surfaces . 10 1.7 Mixed correlation functions and local operator insertion . 13 1.8 Main results from the correspondence . 14 2 Wilson Loops and Supersymmetry 17 2.1 BPS configurations . 17 2.2 Zarembo supersymmetric loops . 18 2.3 DGRT loops . 20 2.3.1 Hopf fibers . 23 2.4 Matrix model . 25 2.5 Calibrated surfaces . 27 3 Strong coupling results 31 3.1 Basic examples . 31 3.1.1 Straight line . 31 3.1.2 Cirle . 32 3.1.3 Antiparallel lines . 33 3.1.4 1/4 BPS circular loop . 33 3.2 Systematic regularization . 35 3.3 Ansatz and excited charges . 36 3.4 Hints of 1-loop computation . 37 v 4 Hopf fibers correlators 41 4.1 Strong coupling solution . 41 4.1.1 S5 motion .