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Essential Concepts of Projective Geomtry Course Notes, MA 561 Purdue University August, 1973 Corrected and supplemented, August, 1978 Reprinted and revised, 2007 Department of Mathematics University of California, Riverside 2007 Table of Contents Preface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : i Prerequisites: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :iv Suggestions for using these notes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :v I. Synthetic and analytic geometry: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :1 1. Axioms for Euclidean geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. Cartesian coordinate interpretations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 3 3. Lines and planes in R and R : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 II. Affine geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1. Synthetic affine geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2. Affine subspaces of vector spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 3. Affine bases: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :19 4. Properties of coordinate affine spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 5. Generalized geometrical incidence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31 6. Isomorphisms and automorphisms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 39 III. Construction of projective space: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :45 1. Ideal points and lines : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 45 2. Homogeneous coordinates : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 51 3. Equations of lines : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 54 4. Higher-dimensional generalizations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 56 Addendum. Synthetic construction of projective space: : : : : : : : : : : : : : :62 IV. Synthetic projective geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 67 1. Axioms for projective geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 67 2. Desargues' Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 71 3. Duality : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 75 4. Conditions for coordinatization: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :84 V. Plane projective geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87 1. Homogeneous line coordinates : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87 2. Cross ratio: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :91 3. Theorems of Desargues and Pappus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 99 4. Complete quadrilaterals and harmonic sets : : : : : : : : : : : : : : : : : : : : : : : : : : 105 5. Interpretation of addition and multiplication : : : : : : : : : : : : : : : : : : : : : : : : 111 VI. Multidimensional projective geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 115 1. Linear varieties and bundles: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :115 2. Projective coordinate systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 120 3. Collineations: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :123 4. Order and separation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 135 VII. Hyperquadrics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 143 1. Definitions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 143 2. Tangents : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 148 3. Bilinear forms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 154 4. Projective classification of hyperquadrics : : : : : : : : : : : : : : : : : : : : : : : : : : : : 158 5. Duality and hyperquadrics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 163 6. Conics in the projective plane : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 165 APPENDICES: A. Review of linear algebra : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 177 1. Vector spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 177 2. Dimensions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 180 3. Systems of linear equations: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :182 4. Linear transformations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 184 5. Dot and cross products : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 188 n Addendum. Rigid motions of R : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 191 B. The join in affine geometry: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :195 C. Reversal of multiplication in skew-fields : : : : : : : : : : : : : : : : : : : : : : : : : : : : 197 D. Algebraic automorphisms of the complex numbers : : : : : : : : : : : : : : : :201 E. Additional material on hyperquadrics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 205 1. Tangent hyperplanes and differential calculus : : : : : : : : : : : : : : : : : : : : : : : : 205 2. Matrices defining the same singular hyperquadric : : : : : : : : : : : : : : : : : : : : 210 Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 215 (219 pages) Electronic file index All files are in the directory http://math.ucr.edu.∼res/progeom pgcontents.pdf The table of contents (this file) pgnotes00.pdf The preface and other preliminaries pgnotes01.pdf Chapter I of the notes pgnotes02.pdf Chapter II of the notes pgnotes03.pdf Chapter III of the notes pgnotes04.pdf Chapter IV of the notes pgnotes05.pdf Chapter V of the notes pgnotes06.pdf Chapter VI of the notes pgnotes07.pdf Chapter VII of the notes pgnotesappa.pdf Appendix A to the notes pgnotesappb.pdf Appendix B to the notes pgnotesappc.pdf Appendix C to the notes pgnotesappd.pdf Appendix D to the notes pgnotesappe.pdf Appendix E to the notes pgnotesbib.pdf The bibliography PREFACE TO THE ORIGINAL 1973 VERSION The purpose of these lecture notes is to present the basic geometrical properties of projec- tive spaces (projective geometry) in a manner reflecting their status in contemporary mathematics. Although projective spaces are no longer studied for their own sake as they were in the early 19th century, they are still a fundamental structure of mathe- matics. Since a wide range of mathematically related careers chosen by undergraduate mathematics students use geometrical ideas1, hopefully these notes will be useful to a correspondingly wide range of students. Of the two basic approaches to projective geometry | the synthetic (in the spirit of classical Euclidean geometry) and the analytic (in the spirit of ordinary analytic geometry) | the latter is generally the more important (and useful) in most modern contexts, and therefore I have tried to emphasize it throughout these notes. Of course, projective geometry (and indeed nearly every subject) should be presented in a relatively efficient manner, and such a treatment of the subject requires the use of both approaches to some extent.2 Thus the synthetic approach is definitely used at the numerous points where it adds significantly to the exposition, and particularly where it helps motivate the translation of geometrical concepts into algebraic data, but the synthetic approach does have a much less dominant position than in many other texts on the subject. We have included a large amount of material from affine geometry in these notes. There are several reasons for this. First of all, one of the basic reasons for studying projective geometry is for its applications to the geometry of Euclidean space, and affine geometry is the fundamental link between projective and Euclidean geometry. Furthermore, a discus- sion of affine geometry allows us to introduce the methods of linear algebra into geometry before projective space is constructed. Each of these is a nontrivial step, and it seems worthwhile to keep them separate. Finally, the linear-algebraic
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  • Lecture 5: Affine Graphics a Connect the Dots Approach to Two-Dimensional Computer Graphics

    Lecture 5: Affine Graphics a Connect the Dots Approach to Two-Dimensional Computer Graphics

    Lecture 5: Affine Graphics A Connect the Dots Approach to Two-Dimensional Computer Graphics The lines are fallen unto me in pleasant places; Psalms 16:6 1. Two Shortcomings of Turtle Graphics Two points determine a line. In Turtle Graphics we use this simple fact to draw a line joining the two points at which the turtle is located before and after the execution of each FORWARD command. By programming the turtle to move about and to draw lines in this fashion, we are able to generate some remarkable figures in the plane. Nevertheless, the turtle has two annoying idiosyncrasies. First, the turtle has no memory, so the order in which the turtle encounters points is crucial. Thus, even though the turtle leaves behind a trace of her path, there is no direct command in LOGO to return the turtle to an arbitrary previously encountered location. Second, the turtle is blissfully unaware of the outside universe. The turtle carries her own local coordinate system -- her state -- but the turtle does not know her position relative to any other point in the plane. Turtle geometry is a local, intrinsic geometry; the turtle knows nothing of the extrinsic, global geometry of the external world. The turtle can draw a circle, but the turtle has no idea where the center of the circle might be or even that there is such a concept as a center, a point outside her path around the circumference. These two shortcomings -- no memory and no knowledge of the outside world -- often make the turtle cumbersome to program.