Using the Hopf Fibration
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Differentiable Manifolds
Prof. A. Cattaneo Institut f¨urMathematik FS 2018 Universit¨atZ¨urich Differentiable Manifolds Solutions to Exercise Sheet 1 p Exercise 1 (A non-differentiable manifold). Consider R with the atlas f(R; id); (R; x 7! sgn(x) x)g. Show R with this atlas is a topological manifold but not a differentiable manifold. p Solution: This follows from the fact that the transition function x 7! sgn(x) x is a homeomor- phism but not differentiable at 0. Exercise 2 (Stereographic projection). Let f : Sn − f(0; :::; 0; 1)g ! Rn be the stereographic projection from N = (0; :::; 0; 1). More precisely, f sends a point p on Sn different from N to the intersection f(p) of the line Np passing through N and p with the equatorial plane xn+1 = 0, as shown in figure 1. Figure 1: Stereographic projection of S2 (a) Find an explicit formula for the stereographic projection map f. (b) Find an explicit formula for the inverse stereographic projection map f −1 (c) If S = −N, U = Sn − N, V = Sn − S and g : Sn ! Rn is the stereographic projection from S, then show that (U; f) and (V; g) form a C1 atlas of Sn. Solution: We show each point separately. (a) Stereographic projection f : Sn − f(0; :::; 0; 1)g ! Rn is given by 1 f(x1; :::; xn+1) = (x1; :::; xn): 1 − xn+1 (b) The inverse stereographic projection f −1 is given by 1 f −1(y1; :::; yn) = (2y1; :::; 2yn; kyk2 − 1): kyk2 + 1 2 Pn i 2 Here kyk = i=1(y ) . -
Understanding Euler Angles 1
Welcome Guest View Cart (0 items) Login Home Products Support News About Us Understanding Euler Angles 1. Introduction Attitude and Heading Sensors from CH Robotics can provide orientation information using both Euler Angles and Quaternions. Compared to quaternions, Euler Angles are simple and intuitive and they lend themselves well to simple analysis and control. On the other hand, Euler Angles are limited by a phenomenon called "Gimbal Lock," which we will investigate in more detail later. In applications where the sensor will never operate near pitch angles of +/‐ 90 degrees, Euler Angles are a good choice. Sensors from CH Robotics that can provide Euler Angle outputs include the GP9 GPS‐Aided AHRS, and the UM7 Orientation Sensor. Figure 1 ‐ The Inertial Frame Euler angles provide a way to represent the 3D orientation of an object using a combination of three rotations about different axes. For convenience, we use multiple coordinate frames to describe the orientation of the sensor, including the "inertial frame," the "vehicle‐1 frame," the "vehicle‐2 frame," and the "body frame." The inertial frame axes are Earth‐fixed, and the body frame axes are aligned with the sensor. The vehicle‐1 and vehicle‐2 are intermediary frames used for convenience when illustrating the sequence of operations that take us from the inertial frame to the body frame of the sensor. It may seem unnecessarily complicated to use four different coordinate frames to describe the orientation of the sensor, but the motivation for doing so will become clear as we proceed. For clarity, this application note assumes that the sensor is mounted to an aircraft. -
Maps That Take Lines to Circles, in Dimension 4
Maps That Take Lines to Circles, in Dimension 4 V. Timorin Abstract. We list all analytic diffeomorphisms between an open subset of the 4-dimen- sional projective space and an open subset of the 4-dimensional sphere that take all line segments to arcs of round circles. These are the following: restrictions of the quaternionic Hopf fibrations and projections from a hyperplane to a sphere from some point. We prove this by finding the exact solutions of the corresponding system of partial differential equations. 1 Introduction Let U be an open subset of the 4-dimensional real projective space RP4 and V an open subset of the 4-dimensional sphere S4. We study diffeomorphisms f : U → V that take all line segments lying in U to arcs of round circles lying in V . For the sake of brevity we will always say in the sequel that f takes all lines to circles. The purpose of this article is to give the complete list of such analytic diffeomorphisms. Remark. Given a diffeomorphism f : U → V that takes lines to circles, we can compose it with a projective transformation in the preimage (which takes lines to lines) and a conformal transformation in the image (which takes circles to circles). The result will be another diffeomorphism taking lines to circles. Example 1. For example, suppose that S4 is embedded in R5 as a Euclidean sphere and take an arbitrary hyperplane and an arbitrary point in R5. Obviously, the pro- jection of the hyperplane to S4 form the given point takes all lines to circles. -
Fixed Point Free Involutions on Cohomology Projective Spaces
Indian J. pure appl. Math., 39(3): 285-291, June 2008 °c Printed in India. FIXED POINT FREE INVOLUTIONS ON COHOMOLOGY PROJECTIVE SPACES HEMANT KUMAR SINGH1 AND TEJ BAHADUR SINGH Department of Mathematics, University of Delhi, Delhi 110 007, India e-mail: tej b [email protected] (Received 22 September 2006; after final revision 24 January 2008; accepted 14 February 2008) Let X be a finitistic space with the mod 2 cohomology ring isomorphic to that of CP n, n odd. In this paper, we determine the mod 2 cohomology ring of the orbit space of a fixed point free involution on X. This gives a classification of cohomology type of spaces with the fundamental group Z2 and the covering space a complex projective space. Moreover, we show that there exists no equivariant map Sm ! X for m > 2 relative to the antipodal action on Sm. An analogous result is obtained for a fixed point free involution on a mod 2 cohomology real projective space. Key Words: Projective space; free action; cohomology algebra; spectral sequence 1. INTRODUCTION Suppose that a topological group G acts (continuously) on a topological space X. Associated with the transformation group (G; X) are two new spaces: The fixed point set XG = fx²Xjgx = x; for all g²Gg and the orbit space X=G whose elements are the orbits G(x) = fgxjg²Gg and the topology is induced by the natural projection ¼ : X ! X=G; x ! G(x). If X and Y are G-spaces, then an equivariant map from X to Y is a continuous map Á : X ! Y such that gÁ(x) = Ág(x) for all g²G; x 2 X. -
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. -
The Hopf Fibration
THE HOPF FIBRATION The Hopf fibration is an important object in fields of mathematics such as topology and Lie groups and has many physical applications such as rigid body mechanics and magnetic monopoles. This project will introduce the Hopf fibration from the points of view of the quaternions and of the complex numbers. n n+1 Consider the standard unit sphere S ⊂ R to be the set of points (x0; x1; : : : ; xn) that satisfy the equation 2 2 2 x0 + x1 + ··· + xn = 1: One way to define the Hopf fibration is via the mapping h : S3 ! S2 given by (1) h(a; b; c; d) = (a2 + b2 − c2 − d2; 2(ad + bc); 2(bd − ac)): You should check that this is indeed a map from S3 to S2. 3 (1) First, we will use the quaternions to study rotations in R . As a set and as a 4 vector space, the set of quaternions is identical to R . There are 3 distinguished coordinate vectors{(0; 1; 0; 0); (0; 0; 1; 0); (0; 0; 0; 1){which are given the names i; j; k respectively. We write the vector (a; b; c; d) as a + bi + cj + dk. The multiplication rules for quaternions can be summarized via the following: i2 = j2 = k2 = −1; ij = k; jk = i; ki = j: Is quaternion multiplication commutative? Is it associative? We can define several other notions associated with quaternions. The conjugate of a quaternionp r = a + bi + cj + dk isr ¯ = a − bi − cj − dk. The norm of r is jjrjj = a2 + b2 + c2 + d2. -
The Real Projective Spaces in Homotopy Type Theory
The real projective spaces in homotopy type theory Ulrik Buchholtz Egbert Rijke Technische Universität Darmstadt Carnegie Mellon University Email: [email protected] Email: [email protected] Abstract—Homotopy type theory is a version of Martin- topology and homotopy theory developed in homotopy Löf type theory taking advantage of its homotopical models. type theory (homotopy groups, including the fundamen- In particular, we can use and construct objects of homotopy tal group of the circle, the Hopf fibration, the Freuden- theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key thal suspension theorem and the van Kampen theorem, players in homotopy theory, as certain higher inductive types for example). Here we give an elementary construction in homotopy type theory. The classical definition of RPn, in homotopy type theory of the real projective spaces as the quotient space identifying antipodal points of the RPn and we develop some of their basic properties. n-sphere, does not translate directly to homotopy type theory. R n In classical homotopy theory the real projective space Instead, we define P by induction on n simultaneously n with its tautological bundle of 2-element sets. As the base RP is either defined as the space of lines through the + case, we take RP−1 to be the empty type. In the inductive origin in Rn 1 or as the quotient by the antipodal action step, we take RPn+1 to be the mapping cone of the projection of the 2-element group on the sphere Sn [4]. -
Evaluation of MEMS Accelerometer and Gyroscope for Orientation Tracking Nutrunner Functionality
EXAMENSARBETE INOM ELEKTROTEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2017 Evaluation of MEMS accelerometer and gyroscope for orientation tracking nutrunner functionality Utvärdering av MEMS accelerometer och gyroskop för rörelseavläsning av skruvdragare ERIK GRAHN KTH SKOLAN FÖR TEKNIK OCH HÄLSA Evaluation of MEMS accelerometer and gyroscope for orientation tracking nutrunner functionality Utvärdering av MEMS accelerometer och gyroskop för rörelseavläsning av skruvdragare Erik Grahn Examensarbete inom Elektroteknik, Grundnivå, 15 hp Handledare på KTH: Torgny Forsberg Examinator: Thomas Lind TRITA-STH 2017:115 KTH Skolan för Teknik och Hälsa 141 57 Huddinge, Sverige Abstract In the production industry, quality control is of importance. Even though today's tools provide a lot of functionality and safety to help the operators in their job, the operators still is responsible for the final quality of the parts. Today the nutrunners manufactured by Atlas Copco use their driver to detect the tightening angle. There- fore the operator can influence the tightening by turning the tool clockwise or counterclockwise during a tightening and quality cannot be assured that the bolt is tightened with a certain torque angle. The function of orientation tracking was de- sired to be evaluated for the Tensor STB angle and STB pistol tools manufactured by Atlas Copco. To be able to study the orientation of a nutrunner, practical exper- iments were introduced where an IMU sensor was fixed on a battery powered nutrunner. Sensor fusion in the form of a complementary filter was evaluated. The result states that the accelerometer could not be used to estimate the angular dis- placement of tightening due to vibration and gimbal lock and therefore a sensor fusion is not possible. -
Dirac's Belt Trick, Gyroscopes, and the Ipad
Dirac's Belt Trick, Gyroscopes, and the iPad Richard Koch May 19, 2012 Dirac's Belt Trick P. A. M. Dirac, 1902 - 1984 Nobel Prize (with Erwin Schrodinger) in 1933 Formulated Dirac equation, a relativistically correct quantum mechanical description of the electron, which predicted the existence of antiparticles. My Home Page: http://pages.uoregon.edu/koch TeXShop for Macintosh; TeX by Donald Knuth for Everything 2 \documentclass[11pt]{amsart} 2 25 Using T X, we can typeset 1+x+x and the matrix . \usepackage[paper width = 6in, paperheight = 7in]{geometry} E e2x+p5 p10 7 \usepackage[parfill]{parskip} According to calculus q ✓ − ◆ \usepackage{graphicx} 1 2 1 x2 p⇡ 2x +3x dx = 2 and e− dx = \begin{document} 2 Z0 Z0 Using \TeX, we can typeset $\sqrt{ {{1 + x + x^2} The path of a particle in a gravitational field is given by γi(t) \over {e^{2x + \sqrt{5}}}}}$ where and the matrix $\left( \begin{array}{cc} 2 & 5 \\ d2γ dγ dγ i + Γi i j =0 \sqrt{10} & -7 \end{array} \right)$. dt2 jk dt dt According to calculus Xjk $$\int_0^1 {2x + 3x^2}\ dx = 2 \hspace{.2in} \mbox{and} \hspace{.2in} \int_0^\infty e^{- x^2} \ dx = {{\sqrt{\pi}} \over 2}$$ The path of a particle in a gravitational field is given by $\gamma_i(t)$ where $${{d^2 \gamma_i} \over {d t^2}} + \sum_{jk} \Gamma^i_{jk} {{d \gamma_i} \over {dt}} {{d \gamma_j} \over {dt}} = 0$$ \begin{figure}[htbp] \centering \includegraphics[width=2in]{MoveTest-math.jpg} \end{figure} \end{document} 1 WWDC, Apple's Worldwide Developer's Conference Gyroscopes in the iPhone and iPad 1. -
Baumgarte Stabilisation Over the SO(3) Rotation Group for Control
2015 IEEE 54th Annual Conference on Decision and Control (CDC) December 15-18, 2015. Osaka, Japan Baumgarte Stabilisation over the SO(3) Rotation Group for Control Sebastien Gros, Marion Zanon, Moritz Diehl Abstract— Representations of the SO(3) rotation group are quaternion and the Direction Cosine Matrix (DCM). For a crucial for airborne and aerospace applications. Euler angles complete survey, we refer to [2]. is a popular representation in many applications, but yield Quaternions are an extension of complex numbers to models having singular dynamics. This issue is addressed via non-singular representations, operating in dimensions higher a four-dimensional space, which allows for describing the than 3. Unit quaternions and the Direction Cosine Matrix are SO(3) rotation group. They can be construed as four- the best known non-singular representations, and favoured in dimensional vectors q R4. The SO(3) rotation group is challenging aeronautic and aerospace applications. All non- 2 4 then covered by q Q := q R q>q = 1 with: singular representations yield invariants in the model dynamics, 2 2 j i.e. a set of nonlinear algebraic conditions that must be fulfilled R(q) = E(q)G(q)> SO(3); by the model initial conditions, and that remain fulfilled over 2 2 3 q1 q0 q3 q2 time. However, due to numerical integration errors, these condi- − − tions tend to become violated when using standard integrators, G(q) = 4 q2 q3 q0 q1 5; − − making the model inconsistent with the physical reality. This q3 q2 q1 q0 issue poses some challenges when non-singular representations 2 − − 3 q1 q0 q3 q2 are deployed in optimal control. -
Example Sheet 1
Part III 2015-16: Differential geometry [email protected] Example Sheet 1 3 1 1. (i) Do the charts '1(x) = x and '2(x) = x (x 2 R) belong to the same C differentiable structure on R? (ii) Let Rj, j = 1; 2, be the manifold defined by using the chart 'j on the topo- logical space R. Are R1 and R2 diffeomorphic? 2. Let X be the metric space defined as follows: Let P1;:::;PN be distinct points in the Euclidean plane with its standard metric d, and define a distance function (not a ∗ 2 ∗ metric for N > 1) ρ on R by ρ (P; Q) = minfd(P; Q); mini;j(d(P; Pi) + d(Pj;Q))g: 2 Let X denote the quotient of R obtained by identifying the N points Pi to a single point P¯ on X. Show that ρ∗ induces a metric on X (the London Underground metric). Show that, for N > 1, the space X cannot be given the structure of a topological manifold. 3. (i) Prove that the product of smooth manifolds has the structure of a smooth manifold. (ii) Prove that n-dimensional real projective space RP n = Sn={±1g has the struc- ture of a smooth manifold of dimension n. (iii) Prove that complex projective space CP n := (Cn+1 nf0g)=C∗ has the structure of a smooth manifold of dimension 2n. 4. (i) Prove that the complex projective line CP 1 is diffeomorphic to the sphere S2. (ii) Show that the natural map (C2 n f0g) ! CP 1 induces a smooth map of manifolds S3 ! S2, the Poincar´emap. -
Geometries of Homogeneous Spaces 1. Rotations of Spheres
(October 9, 2013) Geometries of homogeneous spaces Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/mfms/notes 2013-14/06 homogeneous geometries.pdf] Basic examples of non-Euclidean geometries are best studied by studying the groups that preserve the geometries. Rather than specifying the geometry, we specify the group. The group-invariant geometry on spheres is the familiar spherical geometry, with a simple relation to the ambient Euclidean geometry, also rotation-invariant. The group-invariant geometry on real and complex n-balls is hyperbolic geometry: there are infinitely many straight lines (geodesics) through a given point not on a given straight line, contravening in surplus the parallel postulate for Euclidean geometry. 1. Rotations of spheres 2. Holomorphic rotations n 3. Action of GLn+1(C) on projective space P 4. Real hyperbolic n-space 5. Complex hyperbolic n-space 1. Rotations of spheres Let h; i be the usual inner product on Rn, namely n X hx; yi = xi yi (where x = (x1; : : : ; xn) and y = (y1; : : : ; yn)) i=1 The distance function is expressed in terms of this, as usual: distance from x to y = jx − yj (where jxj = hx; xi1=2) The standard (n − 1)-sphere Sn−1 in Rn is n−1 n S = fx 2 R : jxj = 1g The usual general linear and special linear groups of size n (over R) are 8 < GLn(R) = fn-by-n invertible real matricesg = general linear group : SLn(R) = fg 2 GLn(R) : det g = 1g = special linear group The modifier special refers to the determinant-one condition.