Chapter 6, Analytic Projective Geometry
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
Structural Forms 1
Key principles The hyperboloid of revolution is a Surface and may be generated by revolving a Hyperbola about its conjugate axis. The outline of the elevation will be a Hyperbola. When the conjugate axis is vertical all horizontal sections are circles. The horizontal section at the mid point of the conjugate axis is known as the Throat. The diagram shows the incomplete construction for drawing a hyperbola in a rectangle. (a) Draw the outline of the both branches of the double hyperbola in the rectangle. The diagram shows the incomplete Elevation of a hyperboloid of revolution. (a) Determine the position of the throat circle in elevation. (b) Draw the outline of the both branches of the double hyperbola in elevation. DESIGN & COMMUNICATION GRAPHICS Structural forms 1 NAME: ______________________________ DATE: _____________ The diagram shows the plan and incomplete elevation of an object based on the hyperboloid of revolution. The focal points and transverse axis of the hyperbola are also shown. (a) Using the given information draw the outline of the elevation.. F The diagram shows the axis, focal points and transverse axis of a double hyperbola. (a) Draw the outline of both branches of the double hyperbola. (b) The difference between the focal distances for any point on a double hyperbola is constant and equal to the length of the transverse axis. (c) Indicate this principle on the drawing below. DESIGN & COMMUNICATION GRAPHICS Structural forms 2 NAME: ______________________________ DATE: _____________ Key principles The diagram shows the plan and incomplete elevation of a hyperboloid of revolution. The hyperboloid of revolution may also be generated by revolving one skew line about another. -
Arxiv:2101.02592V1 [Math.HO] 6 Jan 2021 in His Seminal Paper [10]
International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 ©IJCDM Volume 5, 2020, pp. 13{41 Received 6 August 2020. Published on-line 30 September 2020 web: http://www.journal-1.eu/ ©The Author(s) This article is published with open access1. Arrangement of Central Points on the Faces of a Tetrahedron Stanley Rabinowitz 545 Elm St Unit 1, Milford, New Hampshire 03055, USA e-mail: [email protected] web: http://www.StanleyRabinowitz.com/ Abstract. We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 con- ditions occur (such as the four centers being coplanar). A typical result is: The lines from each vertex of a circumscriptible tetrahedron to the Gergonne points of the opposite face are concurrent. Keywords. triangle centers, tetrahedra, computer-discovered mathematics, Eu- clidean geometry. Mathematics Subject Classification (2020). 51M04, 51-08. 1. Introduction Over the centuries, many notable points have been found that are associated with an arbitrary triangle. Familiar examples include: the centroid, the circumcenter, the incenter, and the orthocenter. Of particular interest are those points that Clark Kimberling classifies as \triangle centers". He notes over 100 such points arXiv:2101.02592v1 [math.HO] 6 Jan 2021 in his seminal paper [10]. Given an arbitrary tetrahedron and a choice of triangle center (for example, the circumcenter), we may locate this triangle center in each face of the tetrahedron. -
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. -
Homogeneous Representations of Points, Lines and Planes
Chapter 5 Homogeneous Representations of Points, Lines and Planes 5.1 Homogeneous Vectors and Matrices ................................. 195 5.2 Homogeneous Representations of Points and Lines in 2D ............... 205 n 5.3 Homogeneous Representations in IP ................................ 209 5.4 Homogeneous Representations of 3D Lines ........................... 216 5.5 On Plücker Coordinates for Points, Lines and Planes .................. 221 5.6 The Principle of Duality ........................................... 229 5.7 Conics and Quadrics .............................................. 236 5.8 Normalizations of Homogeneous Vectors ............................. 241 5.9 Canonical Elements of Coordinate Systems ........................... 242 5.10 Exercises ........................................................ 245 This chapter motivates and introduces homogeneous coordinates for representing geo- metric entities. Their name is derived from the homogeneity of the equations they induce. Homogeneous coordinates represent geometric elements in a projective space, as inhomoge- neous coordinates represent geometric entities in Euclidean space. Throughout this book, we will use Cartesian coordinates: inhomogeneous in Euclidean spaces and homogeneous in projective spaces. A short course in the plane demonstrates the usefulness of homogeneous coordinates for constructions, transformations, estimation, and variance propagation. A characteristic feature of projective geometry is the symmetry of relationships between points and lines, called -
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. -
Analytic Representation of Envelope Surfaces Generated by Motion of Surfaces of Revolution Ivana Linkeová
International Journal of Scientific & Engineering Research Volume 8, Issue 8, August-2017 1577 ISSN 2229-5518 Analytic Representation of Envelope Surfaces Generated by Motion of Surfaces of Revolution Ivana Linkeová Abstract—A modified DG/K (Differential Geometry/Kinematics) approach to analytical solution of envelope surfaces generated by continuous motion of a generating surface – general surface of revolution – is presented in this paper. This approach is based on graphical representation of an envelope surface in parametric space of a solid generated by continuous motion of the generating surface. Based on graphical analysis, it is possible to decide whether the envelope surface exists and recognize the expected form of unknown analytical representation of the envelope surface. The obtained results can be used in application of envelope surfaces in mechanical engineering, especially in 3-axis and 5-axis point and flank milling of freeform surfaces. Index Terms—Characteristic curve, Eenvelope surface, Explicit surface, Flank milling, Parametric curve, Parametric surface, Surface of revolution, Tangent plane, Tangent vector. ———————————————————— 1INTRODUCTION ETERMINATION of analytic representation of an enve- analysis of the problem in parametric space of the generated D lope surface generated by continuous motion of a gener- solid. Based on graphical analysis, it is possible to decide ating surface in sufficiently general conceptionof the whether the envelope surface exists or not. If the envelope generating surface as well as the trajectory of motion is a very surface exists, it is possible to recognize expected form of ana- challenging problem of kinematic geometry and its applica- lytical representation of the envelope surface and determine tions in mechanical engineering [1], [2]. -
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. -
Projective Space: Reguli and Projectivity 3
PROJECTIVE SPACE: REGULI AND PROJECTIVITY P.L. ROBINSON Abstract. We investigate an ‘assumption of projectivity’ that is appropriate to the self-dual axiomatic formulation of three-dimensional projective space. 0. Introduction In the traditional Veblen-Young [VY] formulation of projective geometry, based on assump- tions of alignment and extension, it is proved that a projectivity fixing three points on a line fixes all harmonically related points on the line. The Veblen-Young assumptions of alignment and extension are then augmented by a (provisional) ‘assumption of projectivity’ according to which a projectivity that fixes three points on a line fixes all points on the line. See [VY] Chapter IV and Section 35 in particular. In [R] we proposed a self-dual formulation of three-dimensional projective space, founded on lines and abstract incidence, with points and planes as derived notions. The version of the ‘assumption of projectivity’ appropriate to this formulation of projective space should itself be self-dual. Such a version is ready to hand in [VY]: indeed, Veblen and Young note that (with alignment and extension assumed) their ‘assumption of projectivity’ is equivalent to Chapter XI Theorem 1; and this theorem is expressed purely in terms of lines and incidence. Our purpose in this brief note is to indicate some of the results surrounding this self-dual ‘assumption of projectivity’, presenting them within the axiomatic framework of [R]. 1. Framework We recall briefly the axiomatic framework of [R]. The set L of lines is provided with a symmetric reflexive relation of incidence satisfying AXIOM [1] - AXIOM [4] below. For convenience, when S ⊆ L we write S for the set comprising all lines that are incident to each line in S; in case S ={l1,...,ln} we write S = [l1 ...ln]. -
Polarity in Tetrahedral Complexes
Polarity in Tetrahedral Complexes Nick Thomas Introduction Line Geometry, as the name suggests, is a study of the properties of figures formed of lines. In elementary projective geometry the set of coplanar lines in a point, or flat pencil, is well known, as well as the set of tangents to a conic section. Quadric surfaces are of the second order and class in that a line meets one in at most two points and at most two tangent planes lie in that line. In projective geometry there are three types: ellipsoids, ruled quadrics and imaginary quadrics. Only in affine geometry are hyperboloids distinguished from ellipsoids, and only in metric geometry are spheres distinct. Ruled quadrics are generally well known, and may be hyperboloids, paraboloids or cones. They are the next most basic form in line geometry. The abbreviation ªwrtº will be used for ªwith respect toº. Generally lower case latin symbols will be used for lines, upper case latin for points and Greek for planes and parameters. Exotic or italic capitals will be used for figures or configurations. When the term ªgeneralº is used, e.g. 6 general points, this means that special cases such as collinearity of three of them is excluded. Many standard results must be assumed to retain focus and brevity, which if not otherwise referenced will be found in S&K. Line Geometry In spatial geometry figures composed of planes or points are defined by means of a construction or an equation. Those with one degree of freedom possess ∞1 points or planes forming a curve or a developable, while those with two degrees of freedom are surfaces.