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affine geometry

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affine geometry (countable and uncountable, plural affine geometries)

  1. (geometry, uncountable) The branch of geometry dealing with what can be deduced in Euclidean geometry when the notions of line length and angle size are ignored.
    As an alternative to the axiomatic approach, affine geometry can be studied via the properties of affine transformations, which do not, in general, preserve distances or angles, but do preserve alignment of points and parallelism of lines.
    The notion of parallelism remains central to affine geometry, in which the parallel postulate is replaced by Playfair's axiom, a version of the postulate that relies on neither distance nor angle size.
    • 1940 [McGraw-Hill], E. T. Bell, The Development of Mathematics, 2017 [1992], Dover, page 265,
      To include affine geometry, Menger (1935) imposed on lattices a reasonable axiom of parallelism.
    • 1953 [Addison-Wesley], Dirk J. Struik, Lectures on Analytic and Projective Geometry, 2014, Dover, page 108,
      And affine geometry itself can be considered as the geometry of all projective transformations which leave a line invariant.
    • 2005, Miles Reid, Balazs Szendroi, Geometry and Topology, Cambridge University Press, page 62:
      Affine geometry is the geometry of an n-dimensional vector space together with its inhomogeneous linear structure. […] Arbitrary affine linear maps take affine linear subspaces into one another, and also preserve collinearity of points, parallels and ratios of distances along parallel lines; all these are thus well defined notions of affine geometry.
  2. (countable) A geometry that is otherwise Euclidean but disregards lengths and angle sizes.
    • 1989, Walter Prenowitz, Meyer Jordan, Basic Concepts of Geometry, Ardsley House, page 167:
      This chapter is devoted to the theory of affine geometries: those incidence geometries which satisfy the Euclidean parallel postulate in Playfair's form (Ch. 2, Sec. 2).
    • 1992, James G. Oxley, “Matroid Theory”, in Paperback, Oxford University Press, published 2006, page 178:
      The affine geometry   is obtained from  [a projective geometry] by deleting from the latter all the points of a hyperplane.
    • 2001, W. K. Schief, “An Introduction to Integrable Difference and Differential Geometries”, in Alan Coley et al., editors, Bäcklund and Darboux Transformations: The Geometry of Solitons: AARMS-CRM Workshop, American Mathematical Society, page 69:
      We here review work on the discretization of affine geometries which was undertaken in collaboration with A. I. Bobenko [7, 8, 37].

Usage notes edit

Affine geometry is frequently described as what remains of Euclidean geometry when one "forgets" the metric notions of distance and angle.

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