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In mathematics, a reflection (also spelled reflexion)〔"Reflexion" is an archaic spelling.()〕 is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term "reflection" is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a hyperplane. For instance a reflection through a point is an involutive isometry with just one fixed point; the image of the letter p under it would look like a d. This operation is also known as a central inversion , and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane. A figure that does not change upon undergoing a reflection is said to have reflectional symmetry. Some mathematicians use "flip" as a synonym for "reflection".〔 〕〔 〕 ==Construction== In plane (or 3-dimensional) geometry, to find the reflection of a point one drops a perpendicular from the point onto the line (plane) used for reflection, and continues it to the same distance on the other side. To find the reflection of a figure, one reflects each point in the figure. To reflect point P in the line AB using compass and straightedge, proceed as follows (see figure): * Step 1 (red): construct a circle with center at P and some fixed radius ''r'' to create points A′ and B′ on the line AB, which will be equidistant from P. * Step 2 (green): construct circles centered at A′ and B′ having radius ''r''. P and Q will be the points of intersection of these two circles. Point Q is then the reflection of point P in line AB. 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「Reflection (mathematics)」の詳細全文を読む スポンサード リンク
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