16.Transformation Geometry (SC)
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EXAM QUESTIONS (Part Two)
Created by T. Madas MATRICES EXAM QUESTIONS (Part Two) Created by T. Madas Created by T. Madas Question 1 (**) Find the eigenvalues and the corresponding eigenvectors of the following 2× 2 matrix. 7 6 A = . 6 2 2 3 λ= −2, u = α , λ=11, u = β −3 2 Question 2 (**) A transformation in three dimensional space is defined by the following 3× 3 matrix, where x is a scalar constant. 2− 2 4 C =5x − 2 2 . −1 3 x Show that C is non singular for all values of x . FP1-N , proof Created by T. Madas Created by T. Madas Question 3 (**) The 2× 2 matrix A is given below. 1 8 A = . 8− 11 a) Find the eigenvalues of A . b) Determine an eigenvector for each of the corresponding eigenvalues of A . c) Find a 2× 2 matrix P , so that λ1 0 PT AP = , 0 λ2 where λ1 and λ2 are the eigenvalues of A , with λ1< λ 2 . 1 2 1 2 5 5 λ1= −15, λ 2 = 5 , u= , v = , P = − 2 1 − 2 1 5 5 Created by T. Madas Created by T. Madas Question 4 (**) Describe fully the transformation given by the following 3× 3 matrix. 0.28− 0.96 0 0.96 0.28 0 . 0 0 1 rotation in the z axis, anticlockwise, by arcsin() 0.96 Question 5 (**) A transformation in three dimensional space is defined by the following 3× 3 matrix, where k is a scalar constant. 1− 2 k A = k 2 0 . 2 3 1 Show that the transformation defined by A can be inverted for all values of k . -
Two-Dimensional Rotational Kinematics Rigid Bodies
Rigid Bodies A rigid body is an extended object in which the Two-Dimensional Rotational distance between any two points in the object is Kinematics constant in time. Springs or human bodies are non-rigid bodies. 8.01 W10D1 Rotation and Translation Recall: Translational Motion of of Rigid Body the Center of Mass Demonstration: Motion of a thrown baton • Total momentum of system of particles sys total pV= m cm • External force and acceleration of center of mass Translational motion: external force of gravity acts on center of mass sys totaldp totaldVcm total FAext==mm = cm Rotational Motion: object rotates about center of dt dt mass 1 Main Idea: Rotation of Rigid Two-Dimensional Rotation Body Torque produces angular acceleration about center of • Fixed axis rotation: mass Disc is rotating about axis τ total = I α passing through the cm cm cm center of the disc and is perpendicular to the I plane of the disc. cm is the moment of inertial about the center of mass • Plane of motion is fixed: α is the angular acceleration about center of mass cm For straight line motion, bicycle wheel rotates about fixed direction and center of mass is translating Rotational Kinematics Fixed Axis Rotation: Angular for Fixed Axis Rotation Velocity Angle variable θ A point like particle undergoing circular motion at a non-constant speed has SI unit: [rad] dθ ω ≡≡ω kkˆˆ (1)An angular velocity vector Angular velocity dt SI unit: −1 ⎣⎡rad⋅ s ⎦⎤ (2) an angular acceleration vector dθ Vector: ω ≡ Component dt dθ ω ≡ magnitude dt ω >+0, direction kˆ direction ω < 0, direction − kˆ 2 Fixed Axis Rotation: Angular Concept Question: Angular Acceleration Speed 2 ˆˆd θ Object A sits at the outer edge (rim) of a merry-go-round, and Angular acceleration: α ≡≡α kk2 object B sits halfway between the rim and the axis of rotation. -
1.2 Rules for Translations
1.2. Rules for Translations www.ck12.org 1.2 Rules for Translations Here you will learn the different notation used for translations. The figure below shows a pattern of a floor tile. Write the mapping rule for the translation of the two blue floor tiles. Watch This First watch this video to learn about writing rules for translations. MEDIA Click image to the left for more content. CK-12 FoundationChapter10RulesforTranslationsA Then watch this video to see some examples. MEDIA Click image to the left for more content. CK-12 FoundationChapter10RulesforTranslationsB 18 www.ck12.org Chapter 1. Unit 1: Transformations, Congruence and Similarity Guidance In geometry, a transformation is an operation that moves, flips, or changes a shape (called the preimage) to create a new shape (called the image). A translation is a type of transformation that moves each point in a figure the same distance in the same direction. Translations are often referred to as slides. You can describe a translation using words like "moved up 3 and over 5 to the left" or with notation. There are two types of notation to know. T x y 1. One notation looks like (3, 5). This notation tells you to add 3 to the values and add 5 to the values. 2. The second notation is a mapping rule of the form (x,y) → (x−7,y+5). This notation tells you that the x and y coordinates are translated to x − 7 and y + 5. The mapping rule notation is the most common. Example A Sarah describes a translation as point P moving from P(−2,2) to P(1,−1). -
Two-Dimensional Geometric Transformations
Two-Dimensional Geometric Transformations 1 Basic Two-Dimensional Geometric Transformations 2 Matrix Representations and Homogeneous Coordinates 3 Inverse Transformations 4 Two-Dimensional Composite Transformations 5 Other Two-Dimensional Transformations 6 Raster Methods for Geometric Transformations 7 OpenGL Raster Transformations 8 Transformations between Two-Dimensional Coordinate Systems 9 OpenGL Functions for Two-Dimensional Geometric Transformations 10 OpenGL Geometric-Transformation o far, we have seen how we can describe a scene in Programming Examples S 11 Summary terms of graphics primitives, such as line segments and fill areas, and the attributes associated with these primitives. Also, we have explored the scan-line algorithms for displaying output primitives on a raster device. Now, we take a look at transformation operations that we can apply to objects to reposition or resize them. These operations are also used in the viewing routines that convert a world-coordinate scene description to a display for an output device. In addition, they are used in a variety of other applications, such as computer-aided design (CAD) and computer animation. An architect, for example, creates a layout by arranging the orientation and size of the component parts of a design, and a computer animator develops a video sequence by moving the “camera” position or the objects in a scene along specified paths. Operations that are applied to the geometric description of an object to change its position, orientation, or size are called geometric transformations. Sometimes geometric transformations are also referred to as modeling transformations, but some graphics packages make a From Chapter 7 of Computer Graphics with OpenGL®, Fourth Edition, Donald Hearn, M. -
Feature Matching and Heat Flow in Centro-Affine Geometry
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 093, 22 pages Feature Matching and Heat Flow in Centro-Affine Geometry Peter J. OLVER y, Changzheng QU z and Yun YANG x y School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected] URL: http://www.math.umn.edu/~olver/ z School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China E-mail: [email protected] x Department of Mathematics, Northeastern University, Shenyang, 110819, P.R. China E-mail: [email protected] Received April 02, 2020, in final form September 14, 2020; Published online September 29, 2020 https://doi.org/10.3842/SIGMA.2020.093 Abstract. In this paper, we study the differential invariants and the invariant heat flow in centro-affine geometry, proving that the latter is equivalent to the inviscid Burgers' equa- tion. Furthermore, we apply the centro-affine invariants to develop an invariant algorithm to match features of objects appearing in images. We show that the resulting algorithm com- pares favorably with the widely applied scale-invariant feature transform (SIFT), speeded up robust features (SURF), and affine-SIFT (ASIFT) methods. Key words: centro-affine geometry; equivariant moving frames; heat flow; inviscid Burgers' equation; differential invariant; edge matching 2020 Mathematics Subject Classification: 53A15; 53A55 1 Introduction The main objective in this paper is to study differential invariants and invariant curve flows { in particular the heat flow { in centro-affine geometry. In addition, we will present some basic applications to feature matching in camera images of three-dimensional objects, comparing our method with other popular algorithms. -
Multidisciplinary Design Project Engineering Dictionary Version 0.0.2
Multidisciplinary Design Project Engineering Dictionary Version 0.0.2 February 15, 2006 . DRAFT Cambridge-MIT Institute Multidisciplinary Design Project This Dictionary/Glossary of Engineering terms has been compiled to compliment the work developed as part of the Multi-disciplinary Design Project (MDP), which is a programme to develop teaching material and kits to aid the running of mechtronics projects in Universities and Schools. The project is being carried out with support from the Cambridge-MIT Institute undergraduate teaching programe. For more information about the project please visit the MDP website at http://www-mdp.eng.cam.ac.uk or contact Dr. Peter Long Prof. Alex Slocum Cambridge University Engineering Department Massachusetts Institute of Technology Trumpington Street, 77 Massachusetts Ave. Cambridge. Cambridge MA 02139-4307 CB2 1PZ. USA e-mail: [email protected] e-mail: [email protected] tel: +44 (0) 1223 332779 tel: +1 617 253 0012 For information about the CMI initiative please see Cambridge-MIT Institute website :- http://www.cambridge-mit.org CMI CMI, University of Cambridge Massachusetts Institute of Technology 10 Miller’s Yard, 77 Massachusetts Ave. Mill Lane, Cambridge MA 02139-4307 Cambridge. CB2 1RQ. USA tel: +44 (0) 1223 327207 tel. +1 617 253 7732 fax: +44 (0) 1223 765891 fax. +1 617 258 8539 . DRAFT 2 CMI-MDP Programme 1 Introduction This dictionary/glossary has not been developed as a definative work but as a useful reference book for engi- neering students to search when looking for the meaning of a word/phrase. It has been compiled from a number of existing glossaries together with a number of local additions. -
Transformation Geometry — Math 331
Transformation Geometry — Math 331 March 8, 2004 Conjugation of Isometries Definition. If f and g are invertible affine transformations of Rn, the conjugate of f by g is the transformation g ◦ f ◦ g−1. When the context is clear, one sometimes abbreviates notation for conjugation by writing g · f = g ◦ f ◦ g−1 . One views this as g “acting on” f. That is, an affine transformation of Rn not only describes a motion of Rn but also a motion on the set of affine transformations of Rn: the action of the set of affine transformations on itself by conjugation. Theorem 1. If f(x) = Ux + v is an affine transformation of Rn and τ(x) = x + a is translation by the vector a, then f · τ is translation by the vector Ua. Proof. f −1(x) = U −1x − U −1v, and, therefore y = τ ◦ f −1(x) = U −1x − U −1v + a. Then (f · τ)(x) = Uy + v = x + Ua. Warning. Although when a translation is conjugated by an arbitrary affine transformation, the result is a translation, it is not true that the result of conjugating an isometry by an affine transformation is always an isometry. For example, if f is the simple linear transformation (x, y) 7→ (2x, y) of R2 and ρ is rotation through the angle π/2 about the origin, i.e., (x, y) 7→ (−y, x), then f · ρ is the linear map (x, y) 7→ (−2y, x/2), which is not an isometry. On the other hand, it is clear that the result of conjugating an isometry by an isometry is an isometry. -
2-D Drawing Geometry Homogeneous Coordinates
2-D Drawing Geometry Homogeneous Coordinates The rotation of a point, straight line or an entire image on the screen, about a point other than origin, is achieved by first moving the image until the point of rotation occupies the origin, then performing rotation, then finally moving the image to its original position. The moving of an image from one place to another in a straight line is called a translation. A translation may be done by adding or subtracting to each point, the amount, by which picture is required to be shifted. Translation of point by the change of coordinate cannot be combined with other transformation by using simple matrix application. Such a combination is essential if we wish to rotate an image about a point other than origin by translation, rotation again translation. To combine these three transformations into a single transformation, homogeneous coordinates are used. In homogeneous coordinate system, two-dimensional coordinate positions (x, y) are represented by triple- coordinates. Homogeneous coordinates are generally used in design and construction applications. Here we perform translations, rotations, scaling to fit the picture into proper position 2D Transformation in Computer Graphics- In Computer graphics, Transformation is a process of modifying and re- positioning the existing graphics. • 2D Transformations take place in a two dimensional plane. • Transformations are helpful in changing the position, size, orientation, shape etc of the object. Transformation Techniques- In computer graphics, various transformation techniques are- 1. Translation 2. Rotation 3. Scaling 4. Reflection 2D Translation in Computer Graphics- In Computer graphics, 2D Translation is a process of moving an object from one position to another in a two dimensional plane. -
II.3 : Measurement Axioms
I I.3 : Measurement axioms A large — probably dominant — part of elementary Euclidean geometry deals with questions about linear and angular measurements , so it is clear that we must discuss the basic properties of these concepts . Linear measurement In elementary geometry courses , linear measurement is often viewed using the lengths of line segments . We shall view it somewhat differently as giving the (shortest) distance 2 3 between two points . Not surprisingly , in RRR and RRR we define this distance by the usual Pythagorean formula already given in Section I.1. In the synthetic approach , one may view distance as a primitive concept given formally by a function d ; given two points X, Y in the plane or space , the distance d (X, Y) is assumed to be a nonnegative real number , and we further assume that distances have the following standard properties : Axiom D – 1 : The distance d (X, Y) is equal to zero if and only if X = Y. Axiom D – 2 : For all X and Y we have d (X, Y) = d (Y, X). Obviously it is also necessary to have some relationships between an abstract notion of distance and the other undefined concepts introduced thus far ; namely , lines and planes . Intuitively we expect a geometrical line to behave just like the real number line , and the following axiom formulated by G. D. Birkhoff (1884 – 1944) makes this idea precise : Axiom D – 3 ( Ruler Postulate ) : If L is an arbitrary line , then there is a 1 – 1 correspondence between the points of L and the real numbers RRR such that if the points X and Y on L correspond to the real numbers x and y respectively , then we have d (X, Y) = |x – y |. -
Geometric Transformation Techniques for Digital I~Iages: a Survey
GEOMETRIC TRANSFORMATION TECHNIQUES FOR DIGITAL I~IAGES: A SURVEY George Walberg Department of Computer Science Columbia University New York, NY 10027 [email protected] December 1988 Technical Report CUCS-390-88 ABSTRACT This survey presents a wide collection of algorithms for the geometric transformation of digital images. Efficient image transformation algorithms are critically important to the remote sensing, medical imaging, computer vision, and computer graphics communities. We review the growth of this field and compare all the described algorithms. Since this subject is interdisci plinary, emphasis is placed on the unification of the terminology, motivation, and contributions of each technique to yield a single coherent framework. This paper attempts to serve a dual role as a survey and a tutorial. It is comprehensive in scope and detailed in style. The primary focus centers on the three components that comprise all geometric transformations: spatial transformations, resampling, and antialiasing. In addition, considerable attention is directed to the dramatic progress made in the development of separable algorithms. The text is supplemented with numerous examples and an extensive bibliography. This work was supponed in part by NSF grant CDR-84-21402. 6.5.1 Pyramids 68 6.5.2 Summed-Area Tables 70 6.6 FREQUENCY CLAMPING 71 6.7 ANTIALIASED LINES AND TEXT 71 6.8 DISCUSSION 72 SECTION 7 SEPARABLE GEOMETRIC TRANSFORMATION ALGORITHMS 7.1 INTRODUCTION 73 7.1.1 Forward Mapping 73 7.1.2 Inverse Mapping 73 7.1.3 Separable Mapping 74 7.2 2-PASS TRANSFORMS 75 7.2.1 Catmull and Smith, 1980 75 7.2.1.1 First Pass 75 7.2.1.2 Second Pass 75 7.2.1.3 2-Pass Algorithm 77 7.2.1.4 An Example: Rotation 77 7.2.1.5 Bottleneck Problem 78 7.2.1.6 Foldover Problem 79 7.2.2 Fraser, Schowengerdt, and Briggs, 1985 80 7.2.3 Fant, 1986 81 7.2.4 Smith, 1987 82 7.3 ROTATION 83 7.3.1 Braccini and Marino, 1980 83 7.3.2 Weiman, 1980 84 7.3.3 Paeth, 1986/ Tanaka. -
Transformation Groups in Non-Riemannian Geometry Charles Frances
Transformation groups in non-Riemannian geometry Charles Frances To cite this version: Charles Frances. Transformation groups in non-Riemannian geometry. Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics, European Mathematical Society Publishing House, pp.191-216, 2015, 10.4171/148-1/7. hal-03195050 HAL Id: hal-03195050 https://hal.archives-ouvertes.fr/hal-03195050 Submitted on 10 Apr 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Transformation groups in non-Riemannian geometry Charles Frances Laboratoire de Math´ematiques,Universit´eParis-Sud 91405 ORSAY Cedex, France email: [email protected] 2000 Mathematics Subject Classification: 22F50, 53C05, 53C10, 53C50. Keywords: Transformation groups, rigid geometric structures, Cartan geometries. Contents 1 Introduction . 2 2 Rigid geometric structures . 5 2.1 Cartan geometries . 5 2.1.1 Classical examples of Cartan geometries . 7 2.1.2 Rigidity of Cartan geometries . 8 2.2 G-structures . 9 3 The conformal group of a Riemannian manifold . 12 3.1 The theorem of Obata and Ferrand . 12 3.2 Ideas of the proof of the Ferrand-Obata theorem . 13 3.3 Generalizations to rank one parabolic geometries . -
Geometric Transformation of Finite Element Methods: Theory and Applications
GEOMETRIC TRANSFORMATION OF FINITE ELEMENT METHODS: THEORY AND APPLICATIONS MICHAEL HOLST AND MARTIN LICHT Abstract. We present a new technique to apply finite element methods to partial differential equations over curved domains. A change of variables along a coordinate transformation satisfying only low regularity assumptions can translate a Poisson problem over a curved physical domain to a Poisson prob- lem over a polyhedral parametric domain. This greatly simplifies both the geo- metric setting and the practical implementation, at the cost of having globally rough non-trivial coefficients and data in the parametric Poisson problem. Our main result is that a recently developed broken Bramble-Hilbert lemma is key in harnessing regularity in the physical problem to prove higher-order finite el- ement convergence rates for the parametric problem. Numerical experiments are given which confirm the predictions of our theory. 1. Introduction The computational theory of partial differential equations has been in a paradoxi- cal situation from its very inception: partial differential equations over domains with curved boundaries are of theoretical and practical interest, but numerical methods are generally conceived only for polyhedral domains. Overcoming this geomet- ric gap continues to inspire much ongoing research in computational mathematics for treating partial differential equations with geometric features. Computational methods for partial differential equations over curved domains commonly adhere to the philosophy of approximating the physical domain of the partial differential equation by a parametric domain. We mention isoparametric finite element meth- ods [2], surface finite element methods [10, 8, 3], or isogeometric analysis [14] as examples, which describe the parametric domain by a polyhedral mesh whose cells are piecewise distorted to approximate the physical domain closely or exactly.