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Line 5: In [[mathematics]], '''affine geometry''' is what remains of [[Euclidean geometry]] when ignoring (mathematicians often say "forgetting"<ref>{{Citation \| last1=Berger \| first1=Marcel \| author1-link=Marcel Berger \| title=Geometry I \| publisher=Springer \| location=Berlin \| isbn= 3-540-11658-3 \| year=1987}}</ref><ref>See also [[forgetful functor]].</ref>) the [[metric space\|metric]] notions of distance and angle as well as the notion of [[Ordered geometry\|betweenness]] of points on a line. As the notion of ''[[parallel lines]]'' is one of the main properties that is independent of any metric or ordering, affine geometry is often considered as the study of parallel lines. Therefore, [[Playfair's axiom]] (~~''given~~Given a line L and a point P not on L, there is exactly one line parallel to L that passes through P''.) is fundamental in affine geometry. Comparisons of figures in affine geometry are made with [[affine transformation]]s, which are mappings that preserve alignment of points and parallelism of lines. Affine geometry can be developed in two ways that are essentially equivalent.<ref>{{citation \|first=Emil \|last=Artin \|title=[[Geometric Algebra (book)\|Geometric Algebra]] \|series=Wiley Classics Library\|publisher=John Wiley & Sons Inc. \|place=New York \|year=1988 \|pages=x+214 \|isbn=0-471-60839-4\|mr=1009557 \|doi=10.1002/9781118164518}} ''(Reprint of the 1957 original; A Wiley-Interscience Publication)''</ref>

Affine geometry: Difference between revisions