Symmetry and Pattern in Projective Geometry

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  • Projective Geometry.
  • Table of Contents.
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Back to the search result list. Table of Contents Frontmatter Chapter 1.

Projective Geometry 5 Axioms, Duality and Projections

Foundations: The Synthetic Approach Abstract. The axioms for Euclidean, Affine and Projective geometries in two and three dimensions are discussed, with particular emphasis on the status of the theorems of Pappus and Desargues.

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The arithmetic satisfied by the points of a line in affine space is demonstrated and shown to lead to the possibility of affine coordinate systems. After a digression on the symmetry groups of configurations, a set of axioms for Projective geometry of N dimensions is presented. Homogeneous coordinates for a projective plane are introduced and extended to more than two dimensions. This leads to the concept of projective collineations. The theorems of Pappus and Desargues are proved using the method of homogeneous coordinates.

The relation between analytic Projective geometry of N -dimensions and vector space theory is established. The concept of projective collineations is introduced and the classification of collineations in the projective plane described. An important projective invariant, the cross-ratio of four collinear points, is defined.

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Quadric surfaces are introduced and their affine classification is discussed. The theory of the 27 lines on a general cubic surface is presented in detail. Galois fields and the finite Projective geometries that arise from them are introduced. The book is intended for undergraduate students in mathematics and computer science.

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Symmetry and Pattern in Projective Geometry

FAQ Policy. About this Textbook Symmetry and Pattern in Projective Geometry is a self-contained study of projective geometry which compares and contrasts the analytic and axiomatic methods.

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Show all. Linear Figures Pages Lord, Eric. Quadratic Figures Pages Lord, Eric. Cubic Figures Pages Lord, Eric.

Quartic Figures Pages Lord, Eric. Finite Geometries Pages Lord, Eric.