Transformations to Show Congruence and Similarity
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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 . -
Projective Geometry: a Short Introduction
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g. -
Lecture 12: Camera Projection
Robert Collins CSE486, Penn State Lecture 12: Camera Projection Reading: T&V Section 2.4 Robert Collins CSE486, Penn State Imaging Geometry W Object of Interest in World Coordinate System (U,V,W) V U Robert Collins CSE486, Penn State Imaging Geometry Camera Coordinate Y System (X,Y,Z). Z X • Z is optic axis f • Image plane located f units out along optic axis • f is called focal length Robert Collins CSE486, Penn State Imaging Geometry W Y y X Z x V U Forward Projection onto image plane. 3D (X,Y,Z) projected to 2D (x,y) Robert Collins CSE486, Penn State Imaging Geometry W Y y X Z x V u U v Our image gets digitized into pixel coordinates (u,v) Robert Collins CSE486, Penn State Imaging Geometry Camera Image (film) World Coordinates Coordinates CoordinatesW Y y X Z x V u U v Pixel Coordinates Robert Collins CSE486, Penn State Forward Projection World Camera Film Pixel Coords Coords Coords Coords U X x u V Y y v W Z We want a mathematical model to describe how 3D World points get projected into 2D Pixel coordinates. Our goal: describe this sequence of transformations by a big matrix equation! Robert Collins CSE486, Penn State Backward Projection World Camera Film Pixel Coords Coords Coords Coords U X x u V Y y v W Z Note, much of vision concerns trying to derive backward projection equations to recover 3D scene structure from images (via stereo or motion) But first, we have to understand forward projection… Robert Collins CSE486, Penn State Forward Projection World Camera Film Pixel Coords Coords Coords Coords U X x u V Y y v W Z 3D-to-2D Projection • perspective projection We will start here in the middle, since we’ve already talked about this when discussing stereo. -
Point Sets up to Rigid Transformations Are Determined by the Distribution of Their Pairwise Distances
Point Sets Up to Rigid Transformations are Determined by the Distribution of their Pairwise Distances Daniel Chen December 6, 2008 Abstract This report is a summary of [BK06], which gives a simpler, albeit less general proof for the result of [BK04]. 1 Introduction [BK04] and [BK06] study the circumstances when sets of n-points in Euclidean space are determined up to rigid transformations by the distribution of unlabeled pairwise distances. In fact, they show the rather surprising result that any set of n-points in general position are determined up to rigid transformations by the distribution of unlabeled distances. 1.1 Practical Motivation The results of [BK04] and [BK06] are partly motivated by experimental results in [OFCD]. [OFCD] is an experimental study evaluating the use of shape distri- butions to classify 3D objects represented by point clouds. A shape distribution is a the distribution of a function on a randomly selected set of points from the point cloud. [OFCD] tested various such functions, including the angle be- tween three random points on the surface, the distance between the centroid and a random point, the distance between two random points, the square root of the area of the triangle between three random points, and the cube root of the volume of the tetrahedron between four random points. Then, the distri- butions are compared using dissimilarity measures, including χ2, Battacharyya, and Minkowski LN norms for both the pdf and cdf for N = 1; 2; 1. The experimental results showed that the shape distribution function that worked best for classification was the distance between two random points and the best dissimilarity measure was the pdf L1 norm. -
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. -
Draft Framework for a Teaching Unit: Transformations
Co-funded by the European Union PRIMARY TEACHER EDUCATION (PrimTEd) PROJECT GEOMETRY AND MEASUREMENT WORKING GROUP DRAFT FRAMEWORK FOR A TEACHING UNIT Preamble The general aim of this teaching unit is to empower pre-service students by exposing them to geometry and measurement, and the relevant pe dagogical content that would allow them to become skilful and competent mathematics teachers. The depth and scope of the content often go beyond what is required by prescribed school curricula for the Intermediate Phase learners, but should allow pre- service teachers to be well equipped, and approach the teaching of Geometry and Measurement with confidence. Pre-service teachers should essentially be prepared for Intermediate Phase teaching according to the requirements set out in MRTEQ (Minimum Requirements for Teacher Education Qualifications, 2019). “MRTEQ provides a basis for the construction of core curricula Initial Teacher Education (ITE) as well as for Continuing Professional Development (CPD) Programmes that accredited institutions must use in order to develop programmes leading to teacher education qualifications.” [p6]. Competent learning… a mixture of Theoretical & Pure & Extrinsic & Potential & Competent learning represents the acquisition, integration & application of different types of knowledge. Each type implies the mastering of specific related skills Disciplinary Subject matter knowledge & specific specialized subject Pedagogical Knowledge of learners, learning, curriculum & general instructional & assessment strategies & specialized Learning in & from practice – the study of practice using Practical learning case studies, videos & lesson observations to theorize practice & form basis for learning in practice – authentic & simulated classroom environments, i.e. Work-integrated Fundamental Learning to converse in a second official language (LOTL), ability to use ICT & acquisition of academic literacies Knowledge of varied learning situations, contexts & Situational environments of education – classrooms, schools, communities, etc. -
Half-Turns and Line Symmetric Motions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by LSBU Research Open Half-turns and Line Symmetric Motions J.M. Selig Faculty of Business, Computing and Info. Management, London South Bank University, London SE1 0AA, U.K. (e-mail: [email protected]) and M. Husty Institute of Basic Sciences in Engineering Unit Geometry and CAD, Leopold-Franzens-Universit¨at Innsbruck, Austria. (e-mail: [email protected]) Abstract A line symmetric motion is the motion obtained by reflecting a rigid body in the successive generator lines of a ruled surface. In this work we review the dual quaternion approach to rigid body displacements, in particular the representation of the group SE(3) by the Study quadric. Then some classical work on reflections in lines or half-turns is reviewed. Next two new characterisations of line symmetric motions are presented. These are used to study a number of examples one of which is a novel line symmetric motion given by a rational degree five curve in the Study quadric. The rest of the paper investigates the connection between sets of half-turns and linear subspaces of the Study quadric. Line symmetric motions produced by some degenerate ruled surfaces are shown to be restricted to certain 2-planes in the Study quadric. Reflections in the lines of a linear line complex lie in the intersection of a the Study-quadric with a 4-plane. 1 Introduction In this work we revisit the classical idea of half-turns using modern mathematical techniques. In particular we use the dual quaternion representation of rigid-body motions. -
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 . -
Rigid Transformations
Mathematical Principles in Vision and Graphics: Rigid Transformations Ass.Prof. Friedrich Fraundorfer SS 2018 Learning goals . Understand the problems of dealing with rotations . Understand how to represent rotations . Understand the terms SO(3) etc. Understand the use of the tangent space . Understand Euler angles, Axis-Angle, and quaternions . Understand how to interpolate, filter and optimize rotations 2 Outline . Rigid transformations . Problems with rotation matrices . Properties of rotation matrices . Matrix groups SO(3), SE(3) . Manifolds . Tangent space . Skew-symmetric matrices . Exponential map . Euler angles, angle-axis, quaternions . Interpolation . Filtering . Optimization 3 Motivation: 3D Viewer Rigid transformations 푧 푝 푋퐶 푍 푋푤 푥 퐶 푦 푂 푊 푇 푅 푂 푌 푥 ∈ ℝ푛 푋 푋푐 = 푅푋푤 + 푇 T ∈ ℝ푛 . Coordinates are related by: 푋푐 푅 푇 푋푤 = 푛×푛 1 0 1 1 푅 ∈ ℝ . Rigid transformation belong to the matrix group SE(3) . What does this mean? 5 Properties of rotation matrices Rotation matrix: 푧 푍 3×3 푅 = 푟1, 푟2, 푟3 ∈ ℝ 푟3 푦 푟2 푇 푅 푅 = 퐼, det 푅 = +1 푂 푌 푟1 푋 푥 Coordinates are related by: 푋푐 = 푅푋푤 . Rotation matrices belong to the matrix group SO(3) . What does this mean? 6 Problems with rotation matrices . Optimization of rotations (bundle adjustment) 푓(푥푛) ▫ Newton’s method 푥푛+1 = 푥푛 − 푓′(푥푛) . Linear interpolation . Filtering and averaging ▫ E.g. averaging rotation from IMU or camera pose tracker for AR/VR glasses 7 Matrix groups . The set of all the nxn invertible matrices is a group w.r.t. the matrix multiplication: 퐺퐿(푛) = 푀 ∈ ℝ푛×푛|det(푀) ≠ 0 ,× General linear group . -
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.