Solid angle Totally Explained

Everything about The Solid Angle totally explained

The solid angle, Ω, is the angle in three-dimensional space that an object subtends at a point. It is a measure of how big that object appears to an observer looking from that point. For instance, a small object nearby could subtend the same solid angle as a large object far away. The solid angle is proportional to the surface area, S, of a projection of that object onto a sphere centered at that point, divided by the square of the sphere's radius, R. (Symbolically, Ω = k S/R², where k is the proportionality constant.) A solid angle is related to the surface of a sphere in the same way an ordinary angle is related to the circumference of a circle. If the proportionality constant is chosen to be 1, the units of solid angle will be the SI steradian (abbreviated "sr"). Thus 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 sides is one-sixth of that, or 2π/3 sr. Solid angles can also be measured (for k = (180/π)²) in square degrees or (for k = 1/4π) in fractions of the sphere (for example, fractional area).
One way to determine the fractional area subtended by a spherical surface is to divide the area of that surface by the entire surface area of the sphere. The fractional area can then be converted to steradian or square degree measurements by the following formulae:

To obtain the solid angle in steradians, multiply the fractional area by 4π.
To obtain the solid angle in square degrees, multiply the fractional area by 4π × (180/π)², which is equal to 129600/π.

More rigorously, the solid angle for a surface S subtended at a point P is given by the surface integral:
ť

Omega = iint_S frac

This gives the expected results of 2π rad for the 2D circumference and 4π srad for the 3D sphere. It also throws the slightly less obvious 2 for the 1D case, in which the origin-centered unit "sphere" is the interval

[ -1, 1 ]

, which indeed has a measure of 2.

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