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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 Solid angle ： ウィキペディア英語版 Solid angle In geometry, a solid angle (symbol: Ω) is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large the object appears to an observer looking from that point. In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a ''steradian'' (symbol: ''sr''). A small object nearby may subtend the same solid angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Indeed, as viewed from any point on Earth, both objects have approximately the same solid angle as well as apparent size. This is evident during a solar eclipse. == Definition and properties == An object's solid angle in steradians is equal to the area of the segment of a unit sphere, centered at the angle's vertex, that the object covers. A solid angle in steradians equals the area of a segment of a unit sphere in the same way a planar angle in radians equals the length of an arc of a unit circle. Solid angles are often used in physics, in particular astrophysics. The solid angle of an object that is very far away is roughly proportional to the ratio of area to squared distance. Here "area" means the area of the object when projected along the viewing direction. The solid angle of a sphere measured from a point in its interior is 4''π'' sr, and the solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2''π''/3 sr. Solid angles can also be measured in square degrees (1 ''sr'' = (180/''π'')2 ''square degree''), in square minutes and square seconds, or in fractions of the sphere (1 ''sr'' = 1/4''π'' ''fractional area''), also known as spat (1 ''sp'' = 4''π'' ''sr''). In spherical coordinates there is a simple formula for the differential, : $d\backslash Omega\; =\; \backslash sin\backslash theta\backslash ,d\backslash theta\backslash ,d\backslash varphi$ where $\backslash theta$ is the colatitude (angle from the North pole) and $\backslash varphi$ is the longitude. The solid angle for an arbitrary oriented surface S subtended at a point P is equal to the solid angle of the projection of the surface S to the unit sphere with center P, which can be calculated as the surface integral: : $\backslash Omega\; =\; \backslash iint\_S\; \backslash frac\; \backslash ,dS\; \}\backslash \; =\; \backslash iint\_S\; \backslash sin\backslash theta\backslash ,d\backslash theta\backslash ,d\backslash varphi$ where $\backslash hat\; =\; \backslash vec\; /\; r$ is the unit vector corresponding to $\backslash vec$, the position vector of an infinitesimal area of surface $\backslash ,\; dS$ with respect to point P, and where $\backslash hat$ represents the unit normal vector to $\backslash ,\; dS$. Even if the projection on the unit sphere to the surface S is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product $\backslash hat\; \backslash cdot\; \backslash hat$. Thus one can approximate the solid angle subtended by a small facet having flat surface area ''dS'', orientation $\backslash hat$, and distance ''r'' from the viewer as: : $d\backslash Omega\; =\; 4\; \backslash pi\; (dS\; /\; A)\; \backslash ,\; (\backslash hat\; \backslash cdot\; \backslash hat)$ where the surface area of a sphere is $A=4\backslash pi\; r^2$. 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「Solid angle」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.