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In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces that are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. An Euclidean space is an an affine space over the reals, equipped with a metric, the Euclidean distance. Therefore, in Euclidean geometry, an affine property is a property that may be proved in affine spaces. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead ''displacement vectors'', also called ''translation'' vectors or simply ''translations'', between two points of the space.〔The word ''translation'' is generally preferred to ''displacement vector'', which may be confusing, as displacements include also rotations.〕 Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be considered as an affine space, and this amounts to forgetting the special role played by the zero vector. In this case, the elements of the vector space may be viewed either as ''points'' of the affine space or as ''displacement vectors'' or ''translations''. When considered as a point, the zero vector is called the ''origin''. Adding a fixed vector to the elements of a linear subspace of a vector space produces an ''affine subspace''. One commonly says that this affine subspace has been obtained by translating (away of the origin) the linear subspace by the translation vector. In finite dimensions, such an ''affine subspace'' is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding ''homogeneous'' linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space. The dimension of the vector space of the translations, is called the ''dimension of the affine space''. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension in an affine space or a vector space of dimension ''n'' is a affine hyperplane. ==Informal description== The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after you've forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps"). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point — call it — is the origin. Two vectors, and , are to be added. Bob draws an arrow from point to point and another arrow from point to point , and completes the parallelogram to find what Bob thinks is , but Alice knows that he has actually computed :. Similarly, Alice and Bob may evaluate any linear combination of and , or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a linear combination is 1, then Alice and Bob will arrive at the same answer. If Bob travels to : then Alice can similarly travel to :. Then, for all coefficients , Alice and Bob describe the same point with the same linear combination, starting from different origins. While Alice knows the "linear structure", both Alice and Bob know the "affine structure"—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. An underlying set with an affine structure is an affine space. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Affine space」の詳細全文を読む スポンサード リンク
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