Solid angle

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The solid angle, Ω, that an object subtends at a point is a measure of how big that object appears to an observer at 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 area 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 at its center 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 (i.e., 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.

Practical applications

Defining luminous intensity and luminance
Calculating spherical excess E of a spherical triangle
The calculation of potentials by using the Boundary Element Method (BEM)

Solid angles for common objects

An efficient algorithm for calculating the solid angle Ω subtended by a triangle with vertices R₁, R₂ and R₃, as seen from the origin has been given by Oosterom and Strackee (IEEE Trans. Biom. Eng., Vol BME-30, No 2, 1983):

[\tan \left( \frac \Omega \right) = \frac___]}R_R_ + ( _ \cdot _)R_ + ( _ \cdot _)R_ + ( _ \cdot _)R_}],

where:

[R₁R₂R₃] denotes the determinant of the matrix that results when writing the vectors together in a row, e.g. M_ij=R_j(i);

R_i denotes the distance of point i from the origin and R_i is the vector representation of point i;

R_i·R_j denotes the scalar product.

The solid angle of a hemisphere is 2π.

The solid angle of a cone with apex angle [a] is [2 \pi \left (1 - \cos \right)].

The solid angle of a four-sided right regular pyramid with apex angle [a] (measured to the faces of the pyramid) is [4 \arccos \left (-\sin^2 \right) - 2 \pi].

The sun and moon are both seen from Earth at a fractional area of 0.001% of the celestial hemisphere.

Solid Angle in Arbitrary Dimension

The volume of the unit sphere can be defined in any dimension. One often needs this solid angle factor in calculations with spherical symmetry.

[ \Omega_=\frac} \right )}]

Where Γ is the Gamma function.

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