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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Scaling (geometry) ： ウィキペディア英語版 Scaling (geometry) In Euclidean geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc. More general is scaling with a separate scale factor for each axis direction. Non-uniform scaling (anisotropic scaling) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction). Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). It occurs, for example, when a faraway billboard is viewed from an oblique angle, or when the shadow of a flat object falls on a surface that is not parallel to it. When the scale factor is larger than 1, (uniform or non-uniform) scaling is sometimes also called dilation or enlargement. When the scale factor is a positive number smaller than 1, scaling is sometimes also called contraction. In the most general sense, a scaling includes the case in which the directions of scaling are not perpendicular. It also includes the case in which one or more scale factors are equal to zero (projection), and the case of one or more negative scale factors (a directional scaling by -1 is equivalent to a reflection). Scaling is a linear transformation, and a special case of homothetic transformation. In most cases, the homothetic transformations are non-linear transformations. ==Matrix representation== A scaling can be represented by a scaling matrix. To scale an object by a vector ''v'' = (''vx, vy, vz''), each point ''p'' = (''px, py, pz'') would need to be multiplied with this scaling matrix: :$S\_v\; =$ \begin v_x & 0 & 0 \\ 0 & v_y & 0 \\ 0 & 0 & v_z \\ \end. As shown below, the multiplication will give the expected result: :$$ S_vp = \begin v_x & 0 & 0 \\ 0 & v_y & 0 \\ 0 & 0 & v_z \\ \end \begin p_x \\ p_y \\ p_z \end = \begin v_xp_x \\ v_yp_y \\ v_zp_z \end. Such a scaling changes the diameter of an object by a factor between the scale factors, the area by a factor between the smallest and the largest product of two scale factors, and the volume by the product of all three. The scaling is uniform if and only if the scaling factors are equal (''vx = vy = vz''). If all except one of the scale factors are equal to 1, we have directional scaling. In the case where ''vx = vy = vz = k'', scaling increases the area of any surface by a factor of k2 and the volume of any solid object by a factor of k3. 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「Scaling (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.