Line Geometry

Part 1

L.A.D de Boer • Boek • paperback

• Samenvatting
  Line geometry, as its name indicates, studies systems of straight lines in 3-dimensional space. Like with Euclidean Geometry, Line geometry gets a better understanding when
  studied in the extended space, viz. in the real projective space of dimension 3.
  The subject was born, one could say, in 1868, when Julius Plücker finished the first
  volume of his Neue Geometrie des Raumes. It florished in the period around the turn
  of the 19th century into the 20th and was very much embraced by physicists. Though
  it never really disappeared, after WWI it was rather neglected until George Adams
  wrote several studies in the years 1934-1939. In the 1970s Peter Gschwind wrote
  about the Linear Complex and in 1981 Renatus Ziegler gave his first account on line
  geometry. Two interesting books by Stoß followed and in 2012 Ziegler published an
  extended (English) version of his 1981-book. Half of this last book is on general pro-
  jective geometry, and Ziegler deliberately restricted to the synthetic treatment of his
  subjects – as did Stoß. The analytic approach, however, is an important counterpart to the synthetic one, and it is a real joy to discover the differences in proofs between the two approaches.
  By the way, above them reigns algebra. In this book geometric objects are treated as
  algebraic ones, with the fundamental relation of containment (≺) or incidence, and
  the basic operators meet (∧) and join (∨). In chapter 1 this is summarized, as are the
  most important issues of elementary projective geometry. In the next two chapters
  the parabolic strip and the regulus are treated, after which proper line geometry starts with the concept of linear dependency of lines, synthetically as well as analytically.
  Chapter 5 is about linear congruences, chapter 6 about linear complexes. Only then
  it is possible to finish the treatment of dependency of lines.
• Productinformatie
  Binding : Paperback
  Distributievorm : Boek (print, druk)
  Formaat : 155mm x 230mm
  Aantal pagina's : 177
  Uitgeverij : Uitgeverij Geels
  ISBN : 9789083383804
  Datum publicatie : 12-2023
  
• Inhoudsopgave
  Introduction
  1 Recapitulation
  1.1 The general projective space
  1.2 The numeric projective space
  1.3 The projective 3-space over R
  1.4 Degree of Freedom, DoF
  1.5 Separation
  1.6 General position and (in-) dependency
  1.7 Projective maps
  1.8 Conics
  1.9 Coordinates
  1.10 Mutual position of elements
  1.11 Exercises
  1.12 The touch operator
  1.13 Quadrics
  1.14 Projectivities of the line
  1.15 Geometric algebra
  2 The parabolic strip
  2.1 Definition and properties
  2.2 A matrix for the parabolic strip
  2.3 Orientation
  3 The regulus
  3.1 Definition
  3.2 Projectivities
  3.3 Orientation
  4 Linear dependency of lines
  4.1 Synthetic
  4.2 Analytic
  4.3 Outlook
  5 The linear congruence
  5.1 Definitions and properties
  5.2 Lines dependent on 4 independent lines
  5.3 Biaxial collineations
  5.4 Biaxials, synthetic
  5.5 Biaxials, analytic
  5.6 Elations
  5.7 Dependency 4, analytic
  5.8 Orientation of linear congruences
  5.9 Summary
  6 The linear complex
  6.1 The pentagon
  6.2 Special linear complexes
  6.3 An image of a regular linear complex
  6.4 The null-polarity and its complex
  6.5 The theorem of Sylvester
  6.6 The theorem of Chasles
  6.7 Dependency 5, synthetic
  6.8 Constructions
  6.9 Complexes, analytically
  6.10 Dependency: conclusion
  6.11 Orientation of linear complexes
  A.1 The invariant lines of a null-polarity
  A.2 The pencils of a linear complex
  A.3 More about Ω
  B A model of the complex
  List of symbols
  References
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