11 Facts About Affine geometry


In mathematics, affine geometry is what remains of Euclidean geometry when ignoring the metric notions of distance and angle.

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Comparisons of figures in affine geometry are made with affine transformations, which are mappings that preserve alignment of points and parallelism of lines.

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Affine geometry can be developed in two ways that are essentially equivalent.

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Affine geometry used affine geometry to introduce vector addition and subtraction at the earliest stages of his development of mathematical physics.

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The extension to either Euclidean or Minkowskian Affine geometry is achieved by adding various further axioms of orthogonality, etc.

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Euclidean geometry

Various types of affine geometry correspond to what interpretation is taken for rotation.

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An axiomatic treatment of plane affine geometry can be built from the axioms of ordered geometry by the addition of two additional axioms:.

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Since the axioms of ordered geometry as presented here include properties that imply the structure of the real numbers, those properties carry over here so that this is an axiomatization of affine geometry over the field of real numbers.

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Affine geometry can be viewed as the geometry of an affine space of a given dimension n, coordinatized over a field K There is a combinatorial generalization of coordinatized affine space, as developed in synthetic finite geometry.

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In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry.

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Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.

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