Homogeneous Representations of Points, Lines and Planes
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2D and 3D Transformations, Homogeneous Coordinates Lecture 03
2D and 3D Transformations, Homogeneous Coordinates Lecture 03 Patrick Karlsson [email protected] Centre for Image Analysis Uppsala University Computer Graphics November 6 2006 Patrick Karlsson (Uppsala University) Transformations and Homogeneous Coords. Computer Graphics 1 / 23 Reading Instructions Chapters 4.1–4.9. Edward Angel. “Interactive Computer Graphics: A Top-down Approach with OpenGL”, Fourth Edition, Addison-Wesley, 2004. Patrick Karlsson (Uppsala University) Transformations and Homogeneous Coords. Computer Graphics 2 / 23 Todays lecture ... in the pipeline Patrick Karlsson (Uppsala University) Transformations and Homogeneous Coords. Computer Graphics 3 / 23 Scalars, points, and vectors Scalars α, β Real (or complex) numbers. Points P, Q Locations in space (but no size or shape). Vectors u, v Directions in space (magnitude but no position). Patrick Karlsson (Uppsala University) Transformations and Homogeneous Coords. Computer Graphics 4 / 23 Mathematical spaces Scalar field A set of scalars obeying certain properties. New scalars can be formed through addition and multiplication. (Linear) Vector space Made up of scalars and vectors. New vectors can be created through scalar-vector multiplication, and vector-vector addition. Affine space An extended vector space that include points. This gives us additional operators, such as vector-point addition, and point-point subtraction. Patrick Karlsson (Uppsala University) Transformations and Homogeneous Coords. Computer Graphics 5 / 23 Data types Polygon based objects Objects are described using polygons. A polygon is defined by its vertices (i.e., points). Transformations manipulate the vertices, thus manipulates the objects. Some examples in 2D Scalar α 1 float. Point P(x, y) 2 floats. Vector v(x, y) 2 floats. Matrix M 4 floats. -
Finite Projective Geometries 243
FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig. -
Natural Homogeneous Coordinates Edward J
Advanced Review Natural homogeneous coordinates Edward J. Wegman∗ and Yasmin H. Said The natural homogeneous coordinate system is the analog of the Cartesian coordinate system for projective geometry. Roughly speaking a projective geometry adds an axiom that parallel lines meet at a point at infinity. This removes the impediment to line-point duality that is found in traditional Euclidean geometry. The natural homogeneous coordinate system is surprisingly useful in a number of applications including computer graphics and statistical data visualization. In this article, we describe the axioms of projective geometry, introduce the formalism of natural homogeneous coordinates, and illustrate their use with four applications. 2010 John Wiley & Sons, Inc. WIREs Comp Stat 2010 2 678–685 DOI: 10.1002/wics.122 Keywords: projective geometry; crosscap; perspective; parallel coordinates; Lorentz equations PROJECTIVE GEOMETRY Visualizing the projective plane is itself an intriguing exercise. One can imagine an ordinary atural homogeneous coordinates for projective Euclidean plane augmented by a set of points at geometry are the analog of Cartesian coordinates N infinity. Two parallel lines in Euclidean space would for ordinary Euclidean geometry. In two-dimensional meet at a point often called in elementary projective Euclidean geometry, we know that two points will geometry an ideal point. One could imagine that a always determine a line, but the dual statement, two pair of parallel lines would have an ideal point at each lines always determine a point, is not true in general end, i.e. one at −∞ and another at +∞. However, because parallel lines in two dimensions do not meet. there is only one ideal point for a set of parallel lines, The axiomatic framework for projective geometry not one at each end. -
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. -
Projective Geometry Lecture Notes
Projective Geometry Lecture Notes Thomas Baird March 26, 2014 Contents 1 Introduction 2 2 Vector Spaces and Projective Spaces 4 2.1 Vector spaces and their duals . 4 2.1.1 Fields . 4 2.1.2 Vector spaces and subspaces . 5 2.1.3 Matrices . 7 2.1.4 Dual vector spaces . 7 2.2 Projective spaces and homogeneous coordinates . 8 2.2.1 Visualizing projective space . 8 2.2.2 Homogeneous coordinates . 13 2.3 Linear subspaces . 13 2.3.1 Two points determine a line . 14 2.3.2 Two planar lines intersect at a point . 14 2.4 Projective transformations and the Erlangen Program . 15 2.4.1 Erlangen Program . 16 2.4.2 Projective versus linear . 17 2.4.3 Examples of projective transformations . 18 2.4.4 Direct sums . 19 2.4.5 General position . 20 2.4.6 The Cross-Ratio . 22 2.5 Classical Theorems . 23 2.5.1 Desargues' Theorem . 23 2.5.2 Pappus' Theorem . 24 2.6 Duality . 26 3 Quadrics and Conics 28 3.1 Affine algebraic sets . 28 3.2 Projective algebraic sets . 30 3.3 Bilinear and quadratic forms . 31 3.3.1 Quadratic forms . 33 3.3.2 Change of basis . 33 1 3.3.3 Digression on the Hessian . 36 3.4 Quadrics and conics . 37 3.5 Parametrization of the conic . 40 3.5.1 Rational parametrization of the circle . 42 3.6 Polars . 44 3.7 Linear subspaces of quadrics and ruled surfaces . 46 3.8 Pencils of quadrics and degeneration . 47 4 Exterior Algebras 52 4.1 Multilinear algebra . -
F. KLEIN in Leipzig ( *)
“Ueber die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Coordinaten auf eine canonische Form,” Math. Ann 23 (1884), 539-578. On the transformation of the general second-degree equation in line coordinates into a canonical form. By F. KLEIN in Leipzig ( *) Translated by D. H. Delphenich ______ A line complex of degree n encompasses a triply-infinite number of straight lines that are distributed in space in such a manner that those straight lines that go through a fixed point define a cone of order n, or − what says the same thing − that those straight lines that lie in a fixed planes will envelope a curve of class n. Its analytic representation finds such a structure by way of the coordinates of a straight line in space that Pluecker introduced into science ( ** ). According to Pluecker , the straight line has six homogeneous coordinates that fulfill a second-degree condition equation. The straight line will be determined relative to a coordinate tetrahedron by means of it. A homogeneous equation of degree n between these coordinates will represent a complex of degree n. (*) In connection with the republication of some of my older papers in Bd. XXII of these Annals, I am once more publishing my Inaugural Dissertation (Bonn, 1868), a presentation by Lie and myself to the Berlin Academy on Dec. 1870 (see the Monatsberichte), and a note on third-order differential equations that I presented to the sächsischen Gesellschaft der Wissenschaften (last note, with a recently-added Appendix). The Mathematischen Annalen thus contain the totality of my publications up to now, with the single exception of a few that are appearing separately in the book trade, and such provisional publications that were superfluous to later research. -
Chapter 12 the Cross Ratio
Chapter 12 The cross ratio Math 4520, Fall 2017 We have studied the collineations of a projective plane, the automorphisms of the underlying field, the linear functions of Affine geometry, etc. We have been led to these ideas by various problems at hand, but let us step back and take a look at one important point of view of the big picture. 12.1 Klein's Erlanger program In 1872, Felix Klein, one of the leading mathematicians and geometers of his day, in the city of Erlanger, took the following point of view as to what the role of geometry was in mathematics. This is from his \Recent trends in geometric research." Let there be given a manifold and in it a group of transforma- tions; it is our task to investigate those properties of a figure belonging to the manifold that are not changed by the transfor- mation of the group. So our purpose is clear. Choose the group of transformations that you are interested in, and then hunt for the \invariants" that are relevant. This search for invariants has proved very fruitful and useful since the time of Klein for many areas of mathematics, not just classical geometry. In some case the invariants have turned out to be simple polynomials or rational functions, such as the case with the cross ratio. In other cases the invariants were groups themselves, such as homology groups in the case of algebraic topology. 12.2 The projective line In Chapter 11 we saw that the collineations of a projective plane come in two \species," projectivities and field automorphisms. -
Integrability, Normal Forms, and Magnetic Axis Coordinates
INTEGRABILITY, NORMAL FORMS AND MAGNETIC AXIS COORDINATES J. W. BURBY1, N. DUIGNAN2, AND J. D. MEISS2 (1) Los Alamos National Laboratory, Los Alamos, NM 97545 USA (2) Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA Abstract. Integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. Such a field is characterized by the existence of a singular foliation consisting entirely of invariant submanifolds. A regular leaf, known as a flux surface,of this foliation must be diffeomorphic to the two-torus. In a neighborhood of a flux surface, it is known that the magnetic field admits several exact, smooth normal forms in which the field lines are straight. However, these normal forms break down near singular leaves including elliptic and hyperbolic magnetic axes. In this paper, the existence of exact, smooth normal forms for integrable magnetic fields near elliptic and hyperbolic magnetic axes is established. In the elliptic case, smooth near-axis Hamada and Boozer coordinates are defined and constructed. Ultimately, these results establish previously conjectured smoothness properties for smooth solutions of the magnetohydro- dynamic equilibrium equations. The key arguments are a consequence of a geometric reframing of integrability and magnetic fields; that they are presymplectic systems. Contents 1. Introduction 2 2. Summary of the Main Results 3 3. Field-Line Flow as a Presymplectic System 8 3.1. A perspective on the geometry of field-line flow 8 3.2. Presymplectic forms and Hamiltonian flows 10 3.3. Integrable presymplectic systems 13 arXiv:2103.02888v1 [math-ph] 4 Mar 2021 4. -
Geometry of Perspective Imaging Images of the 3-D World
Geometry of perspective imaging ■ Coordinate transformations ■ Image formation ■ Vanishing points ■ Stereo imaging Image formation Images of the 3-D world ■ What is the geometry of the image of a three dimensional object? – Given a point in space, where will we see it in an image? – Given a line segment in space, what does its image look like? – Why do the images of lines that are parallel in space appear to converge to a single point in an image? ■ How can we recover information about the 3-D world from a 2-D image? – Given a point in an image, what can we say about the location of the 3-D point in space? – Are there advantages to having more than one image in recovering 3-D information? – If we know the geometry of a 3-D object, can we locate it in space (say for a robot to pick it up) from a 2-D image? Image formation Euclidean versus projective geometry ■ Euclidean geometry describes shapes “as they are” – properties of objects that are unchanged by rigid motions » lengths » angles » parallelism ■ 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 Image formation Example 1 ■ Consider a set of railroad tracks – Their actual shape: » tracks are parallel » ties are perpendicular to the tracks » ties are evenly spaced along the tracks – Their appearance » tracks converge to a point on the horizon » tracks don’t meet ties at right angles » ties become closer and closer towards the horizon Image formation Example 2 ■ Corner of a room – Actual shape » three walls meeting at right angles. -
Intro to Line Geom and Kinematics
TEUBNER’S MATHEMATICAL GUIDES VOLUME 41 INTRODUCTION TO LINE GEOMETRY AND KINEMATICS BY ERNST AUGUST WEISS ASSOC. PROFESSOR AT THE RHENISH FRIEDRICH-WILHELM-UNIVERSITY IN BONN Translated by D. H. Delphenich 1935 LEIPZIG AND BERLIN PUBLISHED AND PRINTED BY B. G. TEUBNER Foreword According to Felix Klein , line geometry is the geometry of a quadratic manifold in a five-dimensional space. According to Eduard Study , kinematics – viz., the geometry whose spatial element is a motion – is the geometry of a quadratic manifold in a seven- dimensional space, and as such, a natural generalization of line geometry. The geometry of multidimensional spaces is then connected most closely with the geometry of three- dimensional spaces in two different ways. The present guide gives an introduction to line geometry and kinematics on the basis of that coupling. 2 In the treatment of linear complexes in R3, the line continuum is mapped to an M 4 in R5. In that subject, the linear manifolds of complexes are examined, along with the loci of points and planes that are linked to them that lead to their analytic representation, with the help of Weitzenböck’s complex symbolism. One application of the map gives Lie ’s line-sphere transformation. Metric (Euclidian and non-Euclidian) line geometry will be treated, up to the axis surfaces that will appear once more in ray geometry as chains. The conversion principle of ray geometry admits the derivation of a parametric representation of motions from Euler ’s rotation formulas, and thus exhibits the connection between line geometry and kinematics. The main theorem on motions and transfers will be derived by means of the elegant algebra of biquaternions. -
2 Lie Groups
2 Lie Groups Contents 2.1 Algebraic Properties 25 2.2 Topological Properties 27 2.3 Unification of Algebra and Topology 29 2.4 Unexpected Simplification 31 2.5 Conclusion 31 2.6 Problems 32 Lie groups are beautiful, important, and useful because they have one foot in each of the two great divisions of mathematics — algebra and geometry. Their algebraic properties derive from the group axioms. Their geometric properties derive from the identification of group operations with points in a topological space. The rigidity of their structure comes from the continuity requirements of the group composition and inversion maps. In this chapter we present the axioms that define a Lie group. 2.1 Algebraic Properties The algebraic properties of a Lie group originate in the axioms for a group. Definition: A set gi,gj,gk,... (called group elements or group oper- ations) together with a combinatorial operation (called group multi- ◦ plication) form a group G if the following axioms are satisfied: (i) Closure: If g G, g G, then g g G. i ∈ j ∈ i ◦ j ∈ (ii) Associativity: g G, g G, g G, then i ∈ j ∈ k ∈ (g g ) g = g (g g ) i ◦ j ◦ k i ◦ j ◦ k 25 26 Lie Groups (iii) Identity: There is an operator e (the identity operation) with the property that for every group operation g G i ∈ g e = g = e g i ◦ i ◦ i −1 (iv) Inverse: Every group operation gi has an inverse (called gi ) with the property g g−1 = e = g−1 g i ◦ i i ◦ i Example: We consider the set of real 2 2 matrices SL(2; R): × α β A = det(A)= αδ βγ = +1 (2.1) γ δ − where α,β,γ,δ are real numbers. -
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