affine geometry

Noun

affine geometry (countable and uncountable, plural affine geometries)

1. (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.
2. (countable) A geometry that is otherwise Euclidean but disregards lengths and angle sizes.
   • 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).1989, Walter Prenowitz, Meyer Jordan, Basic Concepts of Geometry, Ardsley House, page 167
   • The affine geometry #123;AG#125;(n,F) is obtained from #123;PG#125;(n,F)^([a projective geometry]) by deleting from the latter all the points of a hyperplane.1992, James G. Oxley, “Matroid Theory”, in Paperback, Oxford University Press, published 2006, page 178