Part I, Chapter 3 Simplicial Finite Elements 3.1 Simplices and Barycentric Coordinates
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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. -
Arxiv:1910.10745V1 [Cond-Mat.Str-El] 23 Oct 2019 2.2 Symmetry-Protected Time Crystals
A Brief History of Time Crystals Vedika Khemania,b,∗, Roderich Moessnerc, S. L. Sondhid aDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USA bDepartment of Physics, Stanford University, Stanford, California 94305, USA cMax-Planck-Institut f¨urPhysik komplexer Systeme, 01187 Dresden, Germany dDepartment of Physics, Princeton University, Princeton, New Jersey 08544, USA Abstract The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the per- petuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization (MBL). Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics — that of an intrinsically out-of-equilibrium many-body phase of matter with no equilibrium counterpart. We include a compendium of the necessary background on the statistical mechanics of phase structure in many- body systems, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. In particular, we provide precise definitions that formalize the notion of a time-crystal as a stable, macroscopic, conservative clock — explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. Our discussion emphasizes that TTSB in a time-crystal is accompanied by the breaking of a spatial symmetry — so that time-crystals exhibit a novel form of spatiotemporal order. -
The Fractal Dimension of Product Sets
THE FRACTAL DIMENSION OF PRODUCT SETS A PREPRINT Clayton Moore Williams Brigham Young University [email protected] Machiel van Frankenhuijsen Utah Valley University [email protected] February 26, 2021 ABSTRACT Using methods from nonstandard analysis, we define a nonstandard Minkowski dimension which exists for all bounded sets and which has the property that dim(A × B) = dim(A) + dim(B). That is, our new dimension is “product-summable”. To illustrate our theorem we generalize an example of Falconer’s1 to show that the standard upper Minkowski dimension, as well as the Hausdorff di- mension, are not product-summable. We also include a method for creating sets of arbitrary rational dimension. Introduction There are several notions of dimension used in fractal geometry, which coincide for many sets but have important, distinct properties. Indeed, determining the most proper notion of dimension has been a major problem in geometric measure theory from its inception, and debate over what it means for a set to be fractal has often reduced to debate over the propernotion of dimension. A classical example is the “Devil’s Staircase” set, which is intuitively fractal but which has integer Hausdorff dimension2. As Falconer notes in The Geometry of Fractal Sets3, the Hausdorff dimension is “undoubtedly, the most widely investigated and most widely used” notion of dimension. It can, however, be difficult to compute or even bound (from below) for many sets. For this reason one might want to work with the Minkowski dimension, for which one can often obtain explicit formulas. Moreover, the Minkowski dimension is computed using finite covers and hence has a series of useful identities which can be used in analysis. -
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. -
Paraperspective ≡ Affine
International Journal of Computer Vision, 19(2): 169–180, 1996. Paraperspective ´ Affine Ronen Basri Dept. of Applied Math The Weizmann Institute of Science Rehovot 76100, Israel [email protected] Abstract It is shown that the set of all paraperspective images with arbitrary reference point and the set of all affine images of a 3-D object are identical. Consequently, all uncali- brated paraperspective images of an object can be constructed from a 3-D model of the object by applying an affine transformation to the model, and every affine image of the object represents some uncalibrated paraperspective image of the object. It follows that the paraperspective images of an object can be expressed as linear combinations of any two non-degenerate images of the object. When the image position of the reference point is given the parameters of the affine transformation (and, likewise, the coefficients of the linear combinations) satisfy two quadratic constraints. Conversely, when the values of parameters are given the image position of the reference point is determined by solving a bi-quadratic equation. Key words: affine transformations, calibration, linear combinations, paraperspective projec- tion, 3-D object recognition. 1 Introduction It is shown below that given an object O ½ R3, the set of all images of O obtained by applying a rigid transformation followed by a paraperspective projection with arbitrary reference point and the set of all images of O obtained by applying a 3-D affine transformation followed by an orthographic projection are identical. Consequently, all paraperspective images of an object can be constructed from a 3-D model of the object by applying an affine transformation to the model, and every affine image of the object represents some paraperspective image of the object. -
Fractal-Based Methods As a Technique for Estimating the Intrinsic Dimensionality of High-Dimensional Data: a Survey
INFORMATICA, 2016, Vol. 27, No. 2, 257–281 257 2016 Vilnius University DOI: http://dx.doi.org/10.15388/Informatica.2016.84 Fractal-Based Methods as a Technique for Estimating the Intrinsic Dimensionality of High-Dimensional Data: A Survey Rasa KARBAUSKAITE˙ ∗, Gintautas DZEMYDA Institute of Mathematics and Informatics, Vilnius University Akademijos 4, LT-08663, Vilnius, Lithuania e-mail: [email protected], [email protected] Received: December 2015; accepted: April 2016 Abstract. The estimation of intrinsic dimensionality of high-dimensional data still remains a chal- lenging issue. Various approaches to interpret and estimate the intrinsic dimensionality are deve- loped. Referring to the following two classifications of estimators of the intrinsic dimensionality – local/global estimators and projection techniques/geometric approaches – we focus on the fractal- based methods that are assigned to the global estimators and geometric approaches. The compu- tational aspects of estimating the intrinsic dimensionality of high-dimensional data are the core issue in this paper. The advantages and disadvantages of the fractal-based methods are disclosed and applications of these methods are presented briefly. Key words: high-dimensional data, intrinsic dimensionality, topological dimension, fractal dimension, fractal-based methods, box-counting dimension, information dimension, correlation dimension, packing dimension. 1. Introduction In real applications, we confront with data that are of a very high dimensionality. For ex- ample, in image analysis, each image is described by a large number of pixels of different colour. The analysis of DNA microarray data (Kriukien˙e et al., 2013) deals with a high dimensionality, too. The analysis of high-dimensional data is usually challenging. -
Lecture 16: Planar Homographies Robert Collins CSE486, Penn State Motivation: Points on Planar Surface
Robert Collins CSE486, Penn State Lecture 16: Planar Homographies Robert Collins CSE486, Penn State Motivation: Points on Planar Surface y x Robert Collins CSE486, Penn State Review : Forward Projection World Camera Film Pixel Coords Coords Coords Coords U X x u M M V ext Y proj y Maff v W Z U X Mint u V Y v W Z U M u V m11 m12 m13 m14 v W m21 m22 m23 m24 m31 m31 m33 m34 Robert Collins CSE486, PennWorld State to Camera Transformation PC PW W Y X U R Z C V Rotate to Translate by - C align axes (align origins) PC = R ( PW - C ) = R PW + T Robert Collins CSE486, Penn State Perspective Matrix Equation X (Camera Coordinates) x = f Z Y X y = f x' f 0 0 0 Z Y y' = 0 f 0 0 Z z' 0 0 1 0 1 p = M int ⋅ PC Robert Collins CSE486, Penn State Film to Pixel Coords 2D affine transformation from film coords (x,y) to pixel coordinates (u,v): X u’ a11 a12xa'13 f 0 0 0 Y v’ a21 a22 ya'23 = 0 f 0 0 w’ Z 0 0z1' 0 0 1 0 1 Maff Mproj u = Mint PC = Maff Mproj PC Robert Collins CSE486, Penn StateProjection of Points on Planar Surface Perspective projection y Film coordinates x Point on plane Rotation + Translation Robert Collins CSE486, Penn State Projection of Planar Points Robert Collins CSE486, Penn StateProjection of Planar Points (cont) Homography H (planar projective transformation) Robert Collins CSE486, Penn StateProjection of Planar Points (cont) Homography H (planar projective transformation) Punchline: For planar surfaces, 3D to 2D perspective projection reduces to a 2D to 2D transformation. -
Euclidean Versus Projective Geometry
Projective Geometry Projective Geometry Euclidean versus Projective Geometry n Euclidean geometry describes shapes “as they are” – Properties of objects that are unchanged by rigid motions » Lengths » Angles » Parallelism n Projective geometry describes objects “as they appear” – Lengths, angles, parallelism become “distorted” when we look at objects – Mathematical model for how images of the 3D world are formed. Projective Geometry Overview n Tools of algebraic geometry n Informal description of projective geometry in a plane n Descriptions of lines and points n Points at infinity and line at infinity n Projective transformations, projectivity matrix n Example of application n Special projectivities: affine transforms, similarities, Euclidean transforms n Cross-ratio invariance for points, lines, planes Projective Geometry Tools of Algebraic Geometry 1 n Plane passing through origin and perpendicular to vector n = (a,b,c) is locus of points x = ( x 1 , x 2 , x 3 ) such that n · x = 0 => a x1 + b x2 + c x3 = 0 n Plane through origin is completely defined by (a,b,c) x3 x = (x1, x2 , x3 ) x2 O x1 n = (a,b,c) Projective Geometry Tools of Algebraic Geometry 2 n A vector parallel to intersection of 2 planes ( a , b , c ) and (a',b',c') is obtained by cross-product (a'',b'',c'') = (a,b,c)´(a',b',c') (a'',b'',c'') O (a,b,c) (a',b',c') Projective Geometry Tools of Algebraic Geometry 3 n Plane passing through two points x and x’ is defined by (a,b,c) = x´ x' x = (x1, x2 , x3 ) x'= (x1 ', x2 ', x3 ') O (a,b,c) Projective Geometry Projective Geometry -
Affine Transformations and Rotations
CMSC 425 Dave Mount & Roger Eastman CMSC 425: Lecture 6 Affine Transformations and Rotations Affine Transformations: So far we have been stepping through the basic elements of geometric programming. We have discussed points, vectors, and their operations, and coordinate frames and how to change the representation of points and vectors from one frame to another. Our next topic involves how to map points from one place to another. Suppose you want to draw an animation of a spinning ball. How would you define the function that maps each point on the ball to its position rotated through some given angle? We will consider a limited, but interesting class of transformations, called affine transfor- mations. These include (among others) the following transformations of space: translations, rotations, uniform and nonuniform scalings (stretching the axes by some constant scale fac- tor), reflections (flipping objects about a line) and shearings (which deform squares into parallelograms). They are illustrated in Fig. 1. rotation translation uniform nonuniform reflection shearing scaling scaling Fig. 1: Examples of affine transformations. These transformations all have a number of things in common. For example, they all map lines to lines. Note that some (translation, rotation, reflection) preserve the lengths of line segments and the angles between segments. These are called rigid transformations. Others (like uniform scaling) preserve angles but not lengths. Still others (like nonuniform scaling and shearing) do not preserve angles or lengths. Formal Definition: Formally, an affine transformation is a mapping from one affine space to another (which may be, and in fact usually is, the same space) that preserves affine combi- nations. -
Lecture 04- Projective Geometry
Lecture 04- Projective Geometry EE382-Visual localization & Perception Danping Zou @Shanghai Jiao Tong University 1 Projective geometry • Key concept - Homogenous coordinates – Represent an -dimensional vector by a dimensional coordinate Homogenous coordinate Cartesian coordinate – Can represent infinite points or lines August Ferdinand Möbius Danping Zou @Shanghai Jiao Tong University 1790-1868 2 2D projective geometry • Point representation: – A point can be represented by a homogenous coordinate: Homogeneous coordinates: Inhomogeneous coordinates: Danping Zou @Shanghai Jiao Tong University 3 2D projective geometry • Line representation: – A line is represented by a line equation: Hence a line can be naturally represented by a homogeneous coordinate: It has the same scale equivalence relationship: Danping Zou @Shanghai Jiao Tong University 4 2D projective geometry • A point lie on a line is simply described as: where and . A point or a line has only two degree of freedom (DoF, 自由度). Danping Zou @Shanghai Jiao Tong University 5 2D projective geometry • Intersection of lines: – The intersection of two lines is computed by the cross production of the two homogenous coordinates of the two lines: Intersection of lines Here, ‘ ‘ represents cross production Danping Zou @Shanghai Jiao Tong University 6 2D projective geometry • Example of intersection of two lines: Let the two lines be • The intersection point is computed as : • The inhomogeneous coordinate is Danping Zou @Shanghai Jiao Tong University 7 2D projective geometry • Line across two points: • The line across the two points is obtained by the cross production of the two homogenous coordinates of the two points: Danping Zou @Shanghai Jiao Tong University 8 2D projective geometry • Idea point (point at infinity): – Intersection of parallel lines – Let two parallel lines be where . -
Lecture 6: Definition and Construction of Finite Element Subspaces
18 NUMERICAL SOLUTION OF PDES 2.6. Definition of finite elements. We first define triangular finite element spaces, consid- ering the case when Ω is a convex polygon. For each 0 < h < 1, we let Th be a triangulation of Ω¯ with the following properties: i: Ω¯ = ∪iTi, ii: If Ti and Tj are distinct, then exactly one of the following holds: (a) Ti ∩ Tj = ∅, (b) Ti ∩ Tj is a common vertex, (c) Ti ∩ Tj is a common side. iii: each Ti ∈ Th has diameter ≤ h. B ¢B ¢@ ¢ B PP B ¢ @ B PB¢ B¢ ¢ @ B @ ¢ @ PP B ¢ PB @P¢ PPPBB¢¢ PP PP PP Triangulation of Ω P ¢B PP ¢ B PP ¢@ ¢ B B @ ¢ @ P B @¢ ¢B ¢B PPB @ ¢ B ¢ B @¢ B¢ B @ @ Not allowed @@ @@ @ @ Given a triangulation Th of Ω, a finite element space is a space of piecewise polynomials with respect to Th that is constructed by specifying the following things for each T ∈ Th: Shape functions: a finite dimensional space V (T ) of polynomial functions on T . Degrees of freedom (DOF): a finite set of linear functionals ψi : V (T ) → R (i = 1,..., N) which are unisolvent on V (T ), i.e., if V (T ) has dimension N, then given any numbers αi, i = 1,..., N, there exists a unique function p ∈ V (T ), such that ψi(p) = αi, i = 1,..., N. Since these equations amount to a square linear system of equations, to show unisolvence, we show that the only solution of the system with all αi = 0 is p = 0. A typical example of a DOF is ψi(p) = p(xi). -
Determinants and Transformations 1. (A.) 2
Lecture 3 answers to exercises: Determinants and transformations 1. (a:) 2 · 4 − 3 · (−1) = 11 (b:) − 5 · 2 − 1 · 0 = −10 (c:) Of this matrix we cannot compute the determinant because it is not square. (d:) −5·7·1+1·(−2)·3+(−1)·1·0−(−1)·7·3−1·1·1−(−5)·(−2)·0 = −35−6+21−1 = −21 2. 2 5 −2 4 2 −2 6 4 −1 3 6 −1 1 2 3 −83 3 −1 1 3 42 1 x = = = 20 y = = = −10 4 5 −2 −4 4 4 5 −2 −4 2 3 4 −1 3 4 −1 −1 2 3 −1 2 3 4 5 2 3 4 6 −1 2 1 −57 1 z = = = 14 4 5 −2 −4 4 3 4 −1 −1 2 3 3. Yes. Just think of a matrix and apply it to the zero vector. The outcome of all components are zeros. 4. We find the desired matrix by multiplying the two matrices for the two parts. The 0 −1 matrix for reflection in x + y = 0 is , and the matrix for rotation of 45◦ about −1 0 p p 1 2 − 1 2 2 p 2p the origin is 1 1 . So we compute (note the order!): 2 2 2 2 p p p p 1 2 − 1 2 0 −1 1 2 − 1 2 2 p 2p 2 p 2 p 1 1 = 1 1 2 2 2 2 −1 0 − 2 2 − 2 2 5. A must be the zero matrix. This is true because the vectors 2 3 2, 1 0 2, and 0 2 4 are linearly independent.