DOCSLIB.ORG

2D Geometrical Transformations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
2D Geometrical Transformations 2D Geometrical Transformations Foley & Van Dam, Chapter 5 2D Geometrical Transformations • Translation • Scaling • Rotation • Shear • Matrix notation • Compositions • Homogeneous coordinates 2D Geometrical Transformations Assumption: Objects consist of points and lines. A point is represented by its Cartesian coordinates: P = (x, y) Geometrical Transformation: Let (A, B) be a straight line segment between the points A and B. Let T be a general 2D transformation. T transforms (A, B) into another straight line segment (A’, B’), where: A’=TA and B’=TB Translation • Translate(a, b): (x, y) → (x+a, y+b) Translate(2, 4) Scale • Scale (a, b): (x, y) → (ax, by) Scale (2, 3) Scale (2, 3) Scale • How can we scale an object without moving its origin (lower left corner)? Translate(-1,-1) Translate(1,1) Scale(2,3) Reflection • Special case of scale Scale(-1,1) Scale(1,-1) Rotation • Rotate(θ): (x, y) → (x cos(θ)+y sin(θ), -x sin(θ)+y cos(θ)) Rotate(90) Rotate(90) Rotation • How can we rotate an object without moving its origin (lower left corner)? Translate(-1,-1) Translate(1,1) Rotate(90) Shear • Shear (a, b): (x, y) → (x+ay, y+bx) Shear(1,0) Shear(0,2) Classes of Transformations • Rigid transformation (distance preserving): Translation + Rotation • Similarity transformation (angle preserving): Translation + Rotation + Uniform Scale • Affine transformation (parallelism preserving): Translation + Rotation + Scale + Shear All above transformations Affine are groups where Similarity Rigid ⊂ Similarity ⊂ Affine Rigid Matrix Notation • Let’s treat a point (x, y) as a 2x1 matrix (column vector): x y • What happens when this vector is multiplied by a 2x2 matrix? a b x ax + by = c d y cx + dy 2D Transformations • 2D object is represented by points and lines that join them • Transformations can be applied only to the the points defining the lines • A point (x, y) is represented by a 2x1 column vector, so we can represent 2D transformations by using 2x2 matrices: x ' a b x = y ' c d y Scale • Scale (a, b): (x, y) → (ax, by) a 0 x ax = 0 b y by •If a or b are negative, we get reflection • Inverse: S-1(a,b) = S(1/a, 1/b) Reflection • Reflection through the y axis: − 1 0 0 1 • Reflection through the x axis: 1 0 0 − 1 • Reflection through y = x: 0 1 1 0 • Reflection through y = -x: 0 − 1 − 1 0 Shear • Shear (a, b): (x, y) → (x+ay, y+bx) 1 a x x + ay = b 1 y y + bx Rotation • Rotate(θ): (x, y) → (x cos(θ)+y sin(θ), -x sin(θ)+y cos(θ)) cos θ sin θ x x cos θ + y sin θ = − sin θ cos θ y − x sin θ + y cos θ • Inverse: R-1(q) = RT(q) = R(-q) Composition of Transformations • A sequence of transformations can be collapsed into a single matrix: x x [][][]A B C = [D ] y y •Note: Order of transformations is important! translate rotate rotate translate Translation • Translation (a, b): x x + a → y y + b Problem: Cannot represent translation using 2x2 matrices Solution: Homogeneous Coordinates Homogeneous Coordinates Is a mapping from Rn to Rn+1: ( x , y ) → ( X , Y , W ) = ( tx , ty , t ) Note: All triples (tx, ty, t) correspond to the same non-homogeneous point (x, y) Example (2, 3, 1) ≡ (6, 9, 3). Inverse mapping: X Y ( X , Y , W ) → , W W Translation • Translate(a, b): 1 0 a x x + a 0 1 b y = y + b 0 0 1 1 1 Inverse: T-1(a, b) = T(-a, -b) Affine transformations now have the following form: a b e c d f 0 0 1 Geometric Interpretation A 2D point is mapped to a line (ray) in 3D The non-homogeneous points are obtained by projecting the rays onto the plane Z=1 W Y (X,Y,W) 1 y x (X,Y,1) X Example Rotation about an arbitrary point (x0,y0) θ 1. Translate the coordinates so that the origin is at (x0,y0) 2. Rotate by θ 3. Translate back 1 0 x 0 cos θ sin θ 0 1 0 − x 0 x 0 1 y − sin θ cos θ 0 0 1 − y y = 0 0 0 0 1 0 0 1 0 0 1 1 cos θ − sin θ x 0 (1 − cos θ ) + y 0 sin θ x = sin θ cos θ y (1 − cos θ ) − x sin θ y 0 0 0 0 1 1 Example Reflection about an arbitrary line p2 L = p1 + t (p2-p1) = t p2 + (1-t) p1 p1 1.Translate the coordinates so that P1 is at the origin 2.Rotate so that L aligns with the x-axis 3.Reflect about the x-axis 4.Rotate back 5.Translate back Change of Coordinates It is often required to transform the description of an object from one coordinate system to another Rule other in: Transformthe opposite one direction coordinate of the frame representation towards the change Representation y y’ x’ x Transformation Example • Change of coordinates: Represent P = (xp, yp, 1) in the (x’, y’) coordinate system P ' = MP Where: cos θ sin θ 0 1 0 − x 0 M = − sin θ cos θ 0 0 1 − y 0 0 0 1 0 0 1 y’ x’ y θ (x0, y0) (xp, yp) x Example • Change of coordinates: Alternative method: assume x’ = (ux, uy) and y’ = (vx, vy) in the (x, y) coordinate system where P ' = MP u u 0 1 0 − x x y 0 M = v x v y 0 0 1 − y 0 0 0 1 0 0 1 y’ y x’ (vx, vy) (ux, uy) (x0, y0) x Example Reflection about an arbitrary line p2 L = p1 + t (p2-p1) = t p2 + (1-t) p1 p1 Define a coordinate systems (u, v) parallel to P P : p − p u 1 2 u = 2 1 ≡ x u p 2 − p 1 y − u v v = y = x v u x y 1 0 p u v 0 u u 0 1 0 − p 1x x x 1 0 0 x y 1x M = 0 1 p u v 0 0 − 1 0 v v 0 0 1 − p 1y y y x y 1y 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 3D Viewing Transformation Pipeline Viewing coordinates World Coordinates Object in World ing pp ma :2D 3D 2D:2D mapping Viewport Device Coordinates World to Viewing Coordinates In order to define the viewing window we have to specify: •Windowing-coordinate origin P0 = (x0,y0) •View vector up v = (vx,vy) •Using v, we can find u: u = v x (0,0,1) y v iew (vx,vy). y world (x0,y0)x view (ux,uy). x world Transformation from world to viewing coordinates : u x u y 0 1 0 − x 0 M wc − vc = v x v y 0 0 1 − y 0 0 0 1 0 0 1 Window to Viewport Coordinates (xmax,ymax) (xmin,ymin) Window is Viewing Coordinates Window translated to origin (umaxn,vmaxn) (umin,vmin) Window scaled and translated to Window scaled to Normalized Viewport location in device coordinates Viewport size 1 0 0 1 0 u u u 0 0 x − x 1 0 x min max − min max min − min 1 M vc−dc= 0 1 vmin 0 vmax − vmin 0 0 0 0 1 − ymin ymax − ymin 0 0 1 0 0 1 0 0 1 0 0 1 Normalized Device Coordinates Efficiency Considerations A 2D point transformation requires 9 multiplies and 6 adds ab c x axb++y cz defy=++dx eyfz g hi z g xh++y iz But since affine transformations have always the form: ab c x axb+ y + c de fy=++dx eyf 001 1 1 The number of operations can be reduced to 4 multiplies and 4 adds Efficiency Considerations The rotation matrix is: cos θ sin θ x x cos θ + y sin θ = − sin θ cos θ y − x sin θ + y cos θ When rotating of small angles θ, we can use the fact that cos(θ) ≅ 1 and simplify 1sinθ xxy+ sinθ = −−+sinθθ 1yx sin y Determinant of a Matrix ab = ad− bc cd abc d e f=++−−− aei bfg cdh ceg afh bdi gh i ef df de =−+ab c hi gi gh If P is a polygon of area AP, transformed by a matrix M, the area of the transformed polygon is AP∗|M| .
Load more

Recommended publications
  • Enhancing Self-Reflection and Mathematics Achievement of At-Risk Urban Technical College Students
    Psychological Test and Assessment Modeling, Volume 53, 2011 (1), 108-127 Enhancing self-reflection and mathematics achievement of at-risk urban technical college students Barry J. Zimmerman1, Adam Moylan2, John Hudesman3, Niesha White3, & Bert Flugman3 Abstract A classroom-based intervention study sought to help struggling learners respond to their academic grades in math as sources of self-regulated learning (SRL) rather than as indices of personal limita- tion. Technical college students (N = 496) in developmental (remedial) math or introductory col- lege-level math courses were randomly assigned to receive SRL instruction or conventional in- struction (control) in their respective courses. SRL instruction was hypothesized to improve stu- dents’ math achievement by showing them how to self-reflect (i.e., self-assess and adapt to aca- demic quiz outcomes) more effectively. The results indicated that students receiving self-reflection training outperformed students in the control group on instructor-developed examinations and were better calibrated in their task-specific self-efficacy beliefs before solving problems and in their self- evaluative judgments after solving problems. Self-reflection training also increased students’ pass- rate on a national gateway examination in mathematics by 25% in comparison to that of control students. Key words: self-regulation, self-reflection, math instruction 1 Correspondence concerning this article should be addressed to: Barry Zimmerman, PhD, Graduate Center of the City University of New York and Center for Advanced Study in Education, 365 Fifth Ave- nue, New York, NY 10016, USA; email: [email protected] 2 Now affiliated with the University of California, San Francisco, School of Medicine 3 Graduate Center of the City University of New York and Center for Advanced Study in Education Enhancing self-reflection and math achievement 109 Across America, faculty and policy makers at two-year and technical colleges have been deeply troubled by the low academic achievement and high attrition rate of at-risk stu- dents.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Reflection Invariant and Symmetry Detection
    1 Reflection Invariant and Symmetry Detection Erbo Li and Hua Li Abstract—Symmetry detection and discrimination are of fundamental meaning in science, technology, and engineering. This paper introduces reflection invariants and defines the directional moments(DMs) to detect symmetry for shape analysis and object recognition. And it demonstrates that detection of reflection symmetry can be done in a simple way by solving a trigonometric system derived from the DMs, and discrimination of reflection symmetry can be achieved by application of the reflection invariants in 2D and 3D. Rotation symmetry can also be determined based on that. Also, if none of reflection invariants is equal to zero, then there is no symmetry. And the experiments in 2D and 3D show that all the reflection lines or planes can be deterministically found using DMs up to order six. This result can be used to simplify the efforts of symmetry detection in research areas,such as protein structure, model retrieval, reverse engineering, and machine vision etc. Index Terms—symmetry detection, shape analysis, object recognition, directional moment, moment invariant, isometry, congruent, reflection, chirality, rotation F 1 INTRODUCTION Kazhdan et al. [1] developed a continuous measure and dis- The essence of geometric symmetry is self-evident, which cussed the properties of the reflective symmetry descriptor, can be found everywhere in nature and social lives, as which was expanded to 3D by [2] and was augmented in shown in Figure 1. It is true that we are living in a spatial distribution of the objects asymmetry by [3] . For symmetric world. Pursuing the explanation of symmetry symmetry discrimination [4] defined a symmetry distance will provide better understanding to the surrounding world of shapes.
    [Show full text]
  • Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis
    Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues Meisam Sharify To cite this version: Meisam Sharify. Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues. Numerical Analysis [math.NA]. Ecole Polytechnique X, 2011. English. pastel-00643836 HAL Id: pastel-00643836 https://pastel.archives-ouvertes.fr/pastel-00643836 Submitted on 24 Nov 2011 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. Th`esepr´esent´eepour obtenir le titre de DOCTEUR DE L'ECOLE´ POLYTECHNIQUE Sp´ecialit´e: Math´ematiquesAppliqu´ees par Meisam Sharify Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues Jury Marianne Akian Pr´esident du jury St´ephaneGaubert Directeur Laurence Grammont Examinateur Laura Grigori Examinateur Andrei Sobolevski Rapporteur Fran¸coiseTisseur Rapporteur Paul Van Dooren Examinateur September 2011 Abstract Tropical algebra, which can be considered as a relatively new field in Mathemat- ics, emerged in several branches of science such as optimization, synchronization of production and transportation, discrete event systems, optimal control, oper- ations research, etc.
    [Show full text]
  • 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.
    [Show full text]
  • Efficient Learning of Simplices
    Efficient Learning of Simplices Joseph Anderson Navin Goyal Computer Science and Engineering Microsoft Research India Ohio State University [email protected] [email protected] Luis Rademacher Computer Science and Engineering Ohio State University [email protected] Abstract We show an efficient algorithm for the following problem: Given uniformly random points from an arbitrary n-dimensional simplex, estimate the simplex. The size of the sample and the number of arithmetic operations of our algorithm are polynomial in n. This answers a question of Frieze, Jerrum and Kannan [FJK96]. Our result can also be interpreted as efficiently learning the intersection of n + 1 half-spaces in Rn in the model where the intersection is bounded and we are given polynomially many uniform samples from it. Our proof uses the local search technique from Independent Component Analysis (ICA), also used by [FJK96]. Unlike these previous algorithms, which were based on analyzing the fourth moment, ours is based on the third moment. We also show a direct connection between the problem of learning a simplex and ICA: a simple randomized reduction to ICA from the problem of learning a simplex. The connection is based on a known representation of the uniform measure on a sim- plex. Similar representations lead to a reduction from the problem of learning an affine arXiv:1211.2227v3 [cs.LG] 6 Jun 2013 transformation of an n-dimensional ℓp ball to ICA. 1 Introduction We are given uniformly random samples from an unknown convex body in Rn, how many samples are needed to approximately reconstruct the body? It seems intuitively clear, at least for n = 2, 3, that if we are given sufficiently many such samples then we can reconstruct (or learn) the body with very little error.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • Isometries and the Plane
    Chapter 1 Isometries of the Plane \For geometry, you know, is the gate of science, and the gate is so low and small that one can only enter it as a little child. (W. K. Clifford) The focus of this first chapter is the 2-dimensional real plane R2, in which a point P can be described by its coordinates: 2 P 2 R ;P = (x; y); x 2 R; y 2 R: Alternatively, we can describe P as a complex number by writing P = (x; y) = x + iy 2 C: 2 The plane R comes with a usual distance. If P1 = (x1; y1);P2 = (x2; y2) 2 R2 are two points in the plane, then p 2 2 d(P1;P2) = (x2 − x1) + (y2 − y1) : Note that this is consistent withp the complex notation. For P = x + iy 2 C, p 2 2 recall that jP j = x + y = P P , thus for two complex points P1 = x1 + iy1;P2 = x2 + iy2 2 C, we have q d(P1;P2) = jP2 − P1j = (P2 − P1)(P2 − P1) p 2 2 = j(x2 − x1) + i(y2 − y1)j = (x2 − x1) + (y2 − y1) ; where ( ) denotes the complex conjugation, i.e. x + iy = x − iy. We are now interested in planar transformations (that is, maps from R2 to R2) that preserve distances. 1 2 CHAPTER 1. ISOMETRIES OF THE PLANE Points in the Plane • A point P in the plane is a pair of real numbers P=(x,y). d(0,P)2 = x2+y2. • A point P=(x,y) in the plane can be seen as a complex number x+iy.
    [Show full text]
  • THESIS MULTIDIMENSIONAL SCALING: INFINITE METRIC MEASURE SPACES Submitted by Lara Kassab Department of Mathematics in Partial Fu
    THESIS MULTIDIMENSIONAL SCALING: INFINITE METRIC MEASURE SPACES Submitted by Lara Kassab Department of Mathematics In partial fulfillment of the requirements For the Degree of Master of Science Colorado State University Fort Collins, Colorado Spring 2019 Master’s Committee: Advisor: Henry Adams Michael Kirby Bailey Fosdick Copyright by Lara Kassab 2019 All Rights Reserved ABSTRACT MULTIDIMENSIONAL SCALING: INFINITE METRIC MEASURE SPACES Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces, along with its optimality properties and goodness of fit. This allows us to study the MDS embeddings of the geodesic circle S1 into Rm for all m, and to ask questions about the MDS embeddings of the geodesic n-spheres Sn into Rm. Furthermore, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space X, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of X? Convergence is understood when each metric space in the sequence has the same finite number of points, or when each metric space has a finite number of points tending to infinity. We are also interested in notions of convergence when each metric space in the sequence has an arbitrary (possibly infinite) number of points. ii ACKNOWLEDGEMENTS We would like to thank Henry Adams, Mark Blumstein, Bailey Fosdick, Michael Kirby, Henry Kvinge, Facundo Mémoli, Louis Scharf, the students in Michael Kirby’s Spring 2018 class, and the Pattern Analysis Laboratory at Colorado State University for their helpful conversations and support throughout this project.
    [Show full text]