Solid angle

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The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point. A small object nearby may subtend the same solid angle as a larger object farther away (for example, the small/near Moon can totally eclipse the large/remote Sun because, as observed from a point on the Earth, both objects fill almost exactly the same amount of sky). An object's solid angle is equal to the area of the segment of unit sphere (centered at the vertex of the angle) restricted by the object (this definition works in any dimension, including 1D and 2D). A solid angle equals the area of a segment of unit sphere in the same way a planar angle equals the length of an arc of unit circle.

The units of solid angle can be called steradian (abbreviated "sr") according to SI. From the point of view of mathematics and physics solid angle is dimensionless and has no units, thus "sr" might be skipped in scientific texts. The solid angle of a sphere measured from a point in its interior is 4π sr, and the solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2π/3 sr. Solid angles can also be measured in square degrees (1 sr = (180/π)2 square degree) or in fractions of the sphere (i.e., fractional area), 1 sr = 1/4π fractional area.

In spherical coordinates, there is a simple formula as

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:

where  \vec{r} is the vector position of an infinitesimal area of surface  \, dS with respect to point P and where  \hat{n} represents the unit vector normal to  \, 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 \vec{r} \cdot \hat{n}.


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