Solid angle

In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.
The point from which the object is viewed is called the "apex" of the solid angle, and the object is said to "subtend" its solid angle from that point.
In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a "steradian" (symbol: sr). One steradian corresponds to one unit of area on the unit sphere surrounding the apex, so an object that blocks all rays from the apex would cover a number of steradians equal to the total surface area of the unit sphere, formula_1. Solid angles can also be measured in squares of angular measures such as degrees, minutes, and seconds.
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 apex, 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; therefore, just like a planar angle in radians is the ratio of the length of a circular arc to its radius, a solid angle in steradians is the following ratio:
where A is the spherical surface area and r is the radius of the considered sphere.
Solid angles are often used in astronomy, physics, and 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 any point in its interior is 4 sr, and the solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or  sr. Solid angles can also be measured in square degrees (1 sr = ()2 square degrees), in square minutes and square seconds, or in fractions of the sphere (1 sr = fractional area), also known as spat (1 sp = 4 sr).
In spherical coordinates there is a formula for the differential,
where is the colatitude (angle from the North pole) and 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:
where formula_5 is the unit vector corresponding to formula_6, the position vector of an infinitesimal area of surface with respect to point P, and where formula_7 represents the unit normal vector to . 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 formula_8.
Thus one can approximate the solid angle subtended by a small facet having flat surface area , orientation formula_9, and distance from the viewer as:
where the surface area of a sphere is .
Solid angles for common objects.
Cone, spherical cap, hemisphere.
The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2, is the area of a spherical cap on a unit sphere
formula_11
For small such that , this reduces to the area of a circle .
The above is found by computing the following double integral using the unit surface element in spherical coordinates:
formula_12
This formula can also be derived without the use of calculus. Over 2200 years ago Archimedes proved that the surface area of a spherical cap is always equal to the area of a circle whose radius equals the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap. In the diagram this radius is given as:
formula_13
Hence for a unit sphere the solid angle of the spherical cap is given as:
formula_14
When = , the spherical cap becomes a hemisphere having a solid angle 2.
The solid angle of the complement of the cone is:
formula_15
This is also the solid angle of the part of the celestial sphere that an astronomical observer positioned at latitude can see as the earth rotates. At the equator all of the celestial sphere is visible; at either pole, only one half.
The solid angle subtended by a segment of a spherical cap cut by a plane at angle from the cone's axis and passing through the cone's apex can be calculated by the formula:
formula_16
For example, if , then the formula reduces to the spherical cap formula above: the first term becomes , and the second .
Tetrahedron.
Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where formula_17 are the vector positions of the vertices A, B and C. Define the vertex angle to be the angle BOC and define , correspondingly. Let formula_18 be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define formula_19, formula_20 correspondingly. The solid angle subtended by the triangular surface ABC is given by
This follows from the theory of spherical excess and it leads to the fact that there is an analogous theorem to the theorem that "The sum of internal angles of a planar triangle is equal to ", for the sum of the four internal solid angles of a tetrahedron as follows:
where formula_23 ranges over all six of the dihedral angles between any two planes that contain the tetrahedral faces OAB, OAC, OBC and ABC.
A useful formula for calculating the solid angle of the tetrahedron at the origin O that is purely a function of the vertex angles , , is given by L'Huilier's theorem as
where
Another interesting formula involves expressing the vertices as vectors in 3 dimensional space. Let formula_17 be the vector positions of the vertices A, B and C, and let , , and be the magnitude of each vector (the origin-point distance). The solid angle subtended by the triangular surface ABC is:
where
denotes the scalar triple product of the three vectors and formula_29 denotes the scalar product.
Care must be taken here to avoid negative or incorrect solid angles. One source of potential errors is that the scalar triple product can be negative if , , have the wrong winding. Computing is a sufficient solution since no other portion of the equation depends on the winding. The other pitfall arises when the scalar triple product is positive but the divisor is negative. In this case returns a negative value that must be increased by .
Pyramid.
The solid angle of a four-sided right rectangular pyramid with apex angles and (dihedral angles measured to the opposite side faces of the pyramid) is
If both the side lengths ( and ) of the base of the pyramid and the distance () from the center of the base rectangle to the apex of the pyramid (the center of the sphere) are known, then the above equation can be manipulated to give
The solid angle of a right -gonal pyramid, where the pyramid base is a regular -sided polygon of circumradius , with a 
pyramid height is
The solid angle of an arbitrary pyramid with an -sided base defined by the sequence of unit vectors representing edges can be efficiently computed by:
where parentheses (* *) is a scalar product and square brackets [* * *] is a scalar triple product, and is an imaginary unit. Indices are cycled: and .
Latitude-longitude rectangle.
The solid angle of a latitude-longitude rectangle on a globe is
where and are north and south lines of latitude (measured from the equator in radians with angle increasing northward), and and are east and west lines of longitude (where the angle in radians increases eastward). Mathematically, this represents an arc of angle swept around a sphere by radians. When longitude spans 2 radians and latitude spans radians, the solid angle is that of a sphere.
A latitude-longitude rectangle should not be confused with the solid angle of a rectangular pyramid. All four sides of a rectangular pyramid intersect the sphere's surface in great circle arcs. With a latitude-longitude rectangle, only lines of longitude are great circle arcs; lines of latitude are not.
Sun and Moon.
The Sun is seen from Earth at an average angular diameter of 0.5334 degrees or 9.310 radians. The Moon is seen from Earth at an average angular diameter of 9.22 radians. We can substitute these into the equation given above for the solid angle subtended by a cone with apex angle :
The resulting value for the Sun is 6.807 steradians. The resulting value for the Moon is 6.67 steradians. In terms of the total celestial sphere, the Sun and the Moon subtend "fractional areas" of 0.000542% (5.42 ppm) and 0.000531% (5.31 ppm), respectively. On average, the Sun is larger in the sky than the Moon even though it is much, much farther away.
Solid angles in arbitrary dimensions.
The solid angle subtended by the complete ()-dimensional spherical surface of the unit sphere in -dimensional Euclidean space can be defined in any number of dimensions . One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula
where is the gamma function. When is an integer, the gamma function can be computed explicitly. It follows that
This gives the expected results of 4 steradians for the 3D sphere bounded by a surface of area and 2 radians for the 2D circle bounded by a circumference of length . It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D "sphere" is the interval and this is bounded by two limiting points.
The counterpart to the vector formula in arbitrary dimension was derived by Aomoto
and independently by Ribando. It expresses them as an infinite multivariate Taylor series:
Given unit vectors formula_39 defining the angle, let denote the matrix formed by combining them so the th column is formula_39, and formula_41. The variables formula_42 form a multivariable formula_43. For a "congruent" integer multiexponent formula_44 we define formula_45. 
The notation formula_46 for formula_47 means the variable formula_48, similarly for the exponents formula_49. 
Hence, the term formula_50 means the sum over all terms in formula_51 in which l appears as either the first or second index.
Where this series converges, it converges to the solid angle defined by the vectors.