affine geometry

Definitions

1. The branch of geometry dealing with what can be deduced in Euclidean geometry when the notions of line length and angle size are ignored.
   uncountable
2. A geometry that is otherwise Euclidean but disregards lengths and angle sizes.
   countable

Equivalents

4 more languages

Magyaraffin geometria

日本語アフィン幾何学

Русскийаффи́нная геоме́трия

Svenskaaffin geometri

Examples

“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.”

“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.”

“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).”

“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.”

“We here review⟳ work⟳ on the discretization of affine geometries which was undertaken in collaboration with A. I. Bobenko [7, 8, 37].”

CEFR level