Definitions for affine geometry
affine ge·om·e·t·ry

Princeton's WordNetRate this definition:0.0 / 0 votes

1. affine geometrynoun

   the geometry of affine transformations

WiktionaryRate this definition:0.0 / 0 votes

1. affine geometrynoun

   The branch of geometry dealing with affine transformations.

WikipediaRate this definition:0.0 / 0 votes

1. Affine geometry

   In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. Therefore, 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 affine transformations, which are mappings that preserve alignment of points and parallelism of lines. Affine geometry can be developed in two ways that are essentially equivalent.In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom). Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations. 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 one-to-one correspondence 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). The idea of forgetting the metric can be applied in the theory of manifolds. That is developed in the article on the affine connection.

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1. affine geometry

   Affine geometry is a type of geometry that studies properties of figures which are preserved under affine transformations, which include translations, scaling, shearing, and rotations. Affine geometry does not consider the concepts of angle or length, but focuses on parallel lines, ratios of line segments on a line, and midpoints. It can be seen as an extension of Euclidean geometry that additionally includes the concept of points at infinity.

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1. Affine geometry

   In mathematics, affine geometry is the study of parallel lines. Its use of Playfair's axiom is fundamental since comparative measures of angle size are foreign to affine geometry so that Euclid's parallel postulate is beyond the scope of pure affine geometry. In affine geometry, the relation of parallelism may be adapted so as to be an equivalence relation. Comparisons of figures in affine geometry are made with affinities, which are mappings comprising the affine group A. Since A lies between the Euclidean group E and the group of projectivities P, affine geometry is sometimes mentioned in connection with the Erlangen program, which is concerned with group inclusions such as E ⊂ A ⊂ P. Affine geometry can be developed on the basis of linear algebra. One can define an affine space as a set of points equipped with a set of transformations, the translations, which forms a vector space, and such that for any given ordered pair of points there is a unique translation sending the first point to the second. In more concrete terms, this amounts to having an operation that associates to any two 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. By choosing any point as "origin", the points are in one-to-one correspondence with the vectors, but there is no preferred choice for the origin; thus this approach can be characterized as obtaining the affine space from its associated vector space by "forgetting" the origin.

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1. • Geometry
   • Mathematics

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Numerology

1. Chaldean Numerology

   The numerical value of affine geometry in Chaldean Numerology is: 5

2. Pythagorean Numerology

   The numerical value of affine geometry in Pythagorean Numerology is: 5

References

1. ^ Princeton's WordNet
   http://wordnetweb.princeton.edu/perl/webwn?s=affine geometry
2. ^ Wiktionary
   https://en.wiktionary.org/wiki/Affine_Geometry
3. ^ Wikipedia
   https://en.wikipedia.org/wiki/Affine_Geometry
4. ^ ChatGPT
   https://chat.openai.com
5. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=affine geometry