## Affine geometry | projective view |

geometry aprojecting to asphere .plane outline history

geometers in

,mathematics **affine geometry**is what remains of when not using (mathematicians often say "when forgetting"euclidean geometry ^{[1]}^{[2]}) the notions of distance and angle.metric as the notion of

is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. therefore,parallel lines (playfair's axiom *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 , which are mappings that preserve alignment of points and parallelism of lines.affine transformations affine geometry can be developed in two ways that are essentially equivalent.

^{[3]}in

, ansynthetic geometry is a set ofaffine space *points*to which is associated a set of lines, which satisfy some (such as playfair's axiom).axioms affine geometry can also be developed on the basis of

. in this context anlinear algebra is a set ofaffine space *points*equipped with a set of*transformations*(that is ), the translations, which forms abijective mappings (over a givenvector space , commonly thefield ), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; thereal numbers of two translations is their sum in the vector space of the translations.composition in more concrete terms, this amounts to having an operation that associates to any ordered pair of points a vector and another operation that allows translation of a point by a vector to give another point; these operations are required to satisfy a number of axioms (notably that two successive translations have the effect of translation by the sum vector). by choosing any point as "origin", the points are in

with the vectors, but there is no preferred choice for the origin; thus an affine space may be viewed as obtained from its associated vector space by "forgetting" the origin (zero vector).one-to-one correspondence although this article only discusses

, the notion of "forgetting the metric" is much more general, and can be applied to arbitraryaffine spaces , in general. this extension of the notion of affine spaces to manifolds in general is developed in the article on themanifolds .affine connection - history
- systems of axioms
- affine transformations
- affine space
- projective view
- see also
- references
- further reading
- external links

In traditional ^{[18]} In affine geometry, there is no ^{[19]} In this viewpoint, an

geometry aprojecting to asphere .plane outline history

geometers in

,mathematics **affine geometry**is what remains of when not using (mathematicians often say "when forgetting"euclidean geometry ^{[1]}^{[2]}) the notions of distance and angle.metric as the notion of

is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. therefore,parallel lines (playfair's axiom *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 , which are mappings that preserve alignment of points and parallelism of lines.affine transformations affine geometry can be developed in two ways that are essentially equivalent.

^{[3]}in

, ansynthetic geometry is a set ofaffine space *points*to which is associated a set of lines, which satisfy some (such as playfair's axiom).axioms affine geometry can also be developed on the basis of

. in this context anlinear algebra is a set ofaffine space *points*equipped with a set of*transformations*(that is ), the translations, which forms abijective mappings (over a givenvector space , commonly thefield ), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; thereal numbers of two translations is their sum in the vector space of the translations.composition in more concrete terms, this amounts to having an operation that associates to any ordered pair of points a vector and another operation that allows translation of a point by a vector to give another point; these operations are required to satisfy a number of axioms (notably that two successive translations have the effect of translation by the sum vector). by choosing any point as "origin", the points are in

with the vectors, but there is no preferred choice for the origin; thus an affine space may be viewed as obtained from its associated vector space by "forgetting" the origin (zero vector).one-to-one correspondence although this article only discusses

, the notion of "forgetting the metric" is much more general, and can be applied to arbitraryaffine spaces , in general. this extension of the notion of affine spaces to manifolds in general is developed in the article on themanifolds .affine connection - history
- systems of axioms
- affine transformations
- affine space
- projective view
- see also
- references
- further reading
- external links