Chapter 1 - Projective geometry

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This chapter provides an overview of the standard (unsigned) homogeneous coordinates for the plane, and the classical (unoriented) projective geometry which they implicitly define. It also discusses the advantages and disadvantages of homogeneous coordinates as a computational model, compared to ordinary Cartesian coordinates. The projective plane can be defined either by a means of a concrete model, borrowing concepts from linear algebra or Euclidean geometry, or as an abstract structure satisfying certain axioms. The axiomatic approaches have the advantage of being concise and elegant, but unfortunately they cannot be generalized easily to spaces of arbitrary dimension. To avoid the axiomatic approach and to base all definitions, four concrete models of projective space have been chosen: the straight, spherical, analytic, and vector space models. The use of homogeneous coordinates generally leads to simpler formulas that involve only the basic operations of linear algebra: determinants, dot and cross products, matrix multiplications, and the like. Another advantage of projective geometry is its ability to unify seemingly disparate concepts.