|
''Not to be confused with inversive geometry, in which inversion is through a circle instead of a point.'' In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric. Point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to -1. The point of inversion is also called homothetic center. == Terminology == The term "reflection" is loose, and considered by some an abuse of language, with "inversion" preferred; however, "point reflection" is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identity map – which is also true of other maps called "reflections". More narrowly, a "reflection" refers to a reflection in a hyperplane ( dimensional affine subspace – a point on the line, a line in the plane, a plane in 3-space), with the hyperplane being fixed, but more broadly "reflection" is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension ''k,'' where ) is called the "mirror". In dimension 1 these coincide, as a point is a hyperplane in the line. In terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity ''n''). The term "inversion" should not be confused with inversive geometry, where "inversion" is defined with respect to a circle 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「Point reflection」の詳細全文を読む スポンサード リンク
|