| 翻訳と辞書 |
| Symmetry (geometry) ： ウィキペディア英語版 |
Symmetry (geometry)
A geometric object has ''symmetry'' if there is an "operation" or "transformation" (technically, an isometry or affine map) that maps the figure/object onto itself; i.e., it is said that the object has an invariance under the transform. For instance, a circle rotated about its center will have the same shape and size as the original circle—all points before and after the transform would be indistinguishable. A circle is said to be ''symmetric under rotation'' or to have ''rotational symmetry''. If the isometry is the reflection of a plane figure, the figure is said to have reflectional symmetry or line symmetry; moreover, it is possible for a figure/object to have more than one line of symmetry.
The types of symmetries that are possible for a geometric object depend on the set of geometric transforms available, and on what object properties should remain unchanged after a transform. Because the composition of two transforms is also a transform and every transform has an inverse transform that undoes it, the set of transforms under which an object is symmetric form a mathematical group.
==Symmetries in general==
The most common group of transforms applied to objects are termed the Euclidean group of "isometries," which are distance-preserving transformations in space commonly referred to as two-dimensional or three-dimensional (i.e., in plane geometry or solid geometry Euclidean spaces). These isometries consist of reflections, rotations, translations, and combinations of these basic operations.〔(【引用サイトリンク】 title=Higher Dimensional Group Theory )〕 Under an isometric transformation, a geometric object is said to be symmetric if, after transformation, the object is indistinguishable from the object before the transformation, i.e., if the transformed object is congruent to the original.〔(【引用サイトリンク】url=http://planetmath.org/geometriccongruence )〕 A geometric object is typically symmetric only under a subset or "subgroup" of all isometries. The kinds of isometry subgroups are described below, followed by other kinds of transform groups and by the types of object invariance that are possible in geometry.
Hyperbolic space has another transformation, called striation, parabolic transformation, or pararotation, named after the geological term striation for parallel grooves, and is similar to a Euclidean rotation except the center of rotation is seen at infinity on the ideal limit. Two generating mirrors of a striation create infinitely many virtual copies following a horocycle.
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「Symmetry (geometry)」の詳細全文を読む
スポンサード リンク
| 翻訳と辞書 : 翻訳のためのインターネットリソース |
Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.
|
|