View source for Solid angle

{{short description|Measure of how large an object appears to an observer at a given point in three-dimensional space}}
{{distinguish|spherical angle}}
{{Infobox physical quantity
| name = Solid angle
| width =
| background =
| image = Angle solide coordonnees.svg
| caption = Visual representation of a solid angle
| unit = [[steradian]]
| otherunits = [[Square degree]], [[spat (angular unit)]]
| symbols = Ω
| baseunits = m<sup>2</sup>/m<sup>2</sup>
| dimension = wikidata
| conserved = No
| transformsas =
| derivations = <math>\Omega = A/r^2</math>
}}
In [[geometry]], a '''solid angle''' (symbol: {{math|Ω}}) 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 ''[[Subtended angle|subtend]]'' its solid angle at that point.

In the [[International System of Units]] (SI), a solid angle is expressed in a [[dimensionless quantity|dimensionless]] [[unit of measurement|unit]] called a ''[[steradian]]'' (symbol: sr), which is equal to one square radian, sr = rad<sup>2</sup>. One steradian corresponds to one unit of area (of any shape) 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, <math>4\pi</math>. Solid angles can also be measured in squares of angular measures such as [[Square degree|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 (and therefore apparent size). This is evident during a [[solar eclipse]].

== {{anchor|Square minute|Square second}}Definition and properties ==
{{See also|Spherical polygon area}}
The magnitude of an object's solid angle in [[steradian]]s is equal to the [[area (geometry)|area]] of the segment of a [[unit sphere]], centered at the apex, that the object covers. Giving the area of a segment of a unit sphere in steradians is analogous to giving the length of an arc of a [[unit circle]] in radians. Just as the magnitude of a plane angle in radians at the vertex of a circular sector is the ratio of the length of its arc to its radius, the magnitude of a solid angle in steradians is the ratio of the area covered on a sphere by an object to the square of the radius of the sphere. The formula for the magnitude of the solid angle in steradians is 

<math display=block>\Omega=\frac{A}{r^2}, </math>

where <math>A</math> is the area (of any shape) on the surface of the sphere and <math>r</math> is the radius of the 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.

[[Image:Solid_Angle,_1_Steradian.svg|thumb|Any area on a sphere which is equal in area to the square of its radius, when observed from its center, subtends precisely one [[steradian]].]]

The solid angle of a sphere measured from any point in its interior is 4[[Pi|{{pi}}]]&nbsp;sr. The solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2{{pi}}/3&nbsp; sr. {{anchor|Octant}}The solid angle subtended at the corner of a cube (an [[octant (geometry)|octant]]) or spanned by a [[spherical octant]] is {{pi}}/2&nbsp; sr, one-eight of the solid angle of a sphere.<ref name="u421">{{cite web | title=octant | website=PlanetMath.org | date=2013-03-22 | url=https://planetmath.org/octant | access-date=2024-10-21}}</ref>

Solid angles can also be measured in [[square degree]]s (1&nbsp;sr = {{pars|180/{{pi}}}}<sup>2</sup> square degrees), in square [[arc-minutes]] and square [[arc-seconds]], or in fractions of the sphere (1&nbsp;sr = {{sfrac|1|4{{pi}}}} fractional area), also known as [[spat (angular unit)|spat]] (1&nbsp;sp = 4{{pi}}&nbsp;sr).

In [[spherical coordinates#Integration_and_differentiation_in_spherical_coordinates|spherical coordinates]] there is a formula for the [[Differential of a function|differential]],

<math display=block>d\Omega = \sin\theta\,d\theta\,d\varphi,</math>

where {{mvar|θ}} is the [[colatitude]] (angle from the North Pole) and {{mvar|φ}} is the longitude.

The solid angle for an arbitrary [[oriented surface]] {{mvar|S}} subtended at a point {{mvar|P}} is equal to the solid angle of the projection of the surface {{mvar|S}} to the unit sphere with center {{mvar|P}}, which can be calculated as the [[surface integral]]:

<math display=block>\Omega = \iint_S \frac{ \hat{r} \cdot \hat{n}}{r^2}\,dS \ = \iint_S \sin\theta\,d\theta\,d\varphi,</math>

where <math>\hat{r} = \vec{r} / r</math> is the [[unit vector]] corresponding to <math> \vec{r} </math>, the [[position vector]] of an infinitesimal area of surface {{math|''dS''}} with respect to point {{mvar|P}}, and where <math> \hat{n} </math> represents the unit [[normal vector]] to {{math|''dS''}}. Even if the projection on the unit sphere to the surface {{mvar|S}} is not [[isomorphic]], the multiple folds are correctly considered according to the surface orientation described by the sign of the [[scalar product]] <math>\hat{r} \cdot \hat{n}</math>.

Thus one can approximate the solid angle subtended by a small [[facet]] having flat surface area {{math|''dS''}}, orientation <math>\hat{n}</math>, and distance {{math|''r''}} from the viewer as:

<math display=block>d\Omega = 4 \pi \left(\frac{dS}{A}\right) \, (\hat{r} \cdot \hat{n}),</math>

where the [[surface area of a sphere]] is {{math|1=''A'' = 4{{pi}}''r''<sup>2</sup>}}.

==Practical applications==
*Defining [[luminous intensity]] and [[luminance]], and the correspondent radiometric quantities [[radiant intensity]] and [[radiance]]
*Calculating [[Spherical trigonometry#Area and spherical excess|spherical excess]] {{math|''E''}} of a [[spherical triangle]]
*The calculation of potentials by using the [[boundary element method]] (BEM)
*Evaluating the size of [[ligand]]s in metal complexes, see [[ligand cone angle]]
*Calculating the [[electric field]] and [[magnetic field]] strength around charge distributions
*Deriving [[Gauss's Law]]
*Calculating emissive power and irradiation in heat transfer
*Calculating cross sections in [[Rutherford scattering]]
*Calculating cross sections in [[Raman scattering]]
*The solid angle of the [[acceptance cone]] of the [[optical fiber]]
*The computation of nodal densities in meshes.<ref>{{cite journal| last = Falla| first = Romain| year = 2023| title = Mesh adaption for two-dimensional bounded and free-surface flows with the particle finite element method| journal = Computational Particle Mechanics| volume = 10| issue = 5| pages = 1049–1076| doi = 10.1007/s40571-022-00541-2| bibcode = 2023CPM....10.1049F| url = https://link.springer.com/article/10.1007/s40571-022-00541-2}}</ref>

== Solid angles for common objects ==

=== Cone, spherical cap, hemisphere  ===
[[Image:Steradian cone and cap.svg|thumb|right|250px|Diagram showing a section through the centre of a cone (1) subtending a solid angle of 1 steradian in a sphere of radius r, along with the spherical "cap" (2). The external surface area A of the cap equals r2 only if solid angle of the cone is exactly 1 steradian. Hence, in this figure {{math|''θ'' {{=}} ''A''/2}} and {{math|''r'' {{=}} 1}}.]]
The solid angle of a [[cone (geometry)|cone]] with its apex at the apex of the solid angle, and with [[Apex (geometry)|apex]] angle 2{{math|''θ''}}, is the area of a [[spherical cap]] on a [[unit sphere]]

<math display=block>\Omega = 2\pi \left (1 - \cos\theta \right)\ = 4\pi \sin^2 \frac{\theta}{2}.</math>

For small {{math|''θ''}} such that {{math|cos ''θ'' ≈ 1 − {{sfrac|''θ''<sup>2</sup>|2}}}} this reduces to {{math|π''θ''<sup>2</sup>}}, the area of a circle.

The above is found by computing the following [[double integral]] using the unit [[Spherical coordinate system#Integration and differentiation in spherical coordinates|surface element in spherical coordinates]]:

<math display=block>\begin{align}
\int_0^{2\pi} \int_0^\theta \sin\theta' \, d \theta' \, d \phi
&= \int_0^{2\pi} d \phi\int_0^\theta \sin\theta' \, d \theta' \\
&= 2\pi\int_0^\theta \sin\theta' \, d \theta' \\
&= 2\pi\left[ -\cos\theta' \right]_0^{\theta} \\
&= 2\pi\left(1 - \cos\theta \right).
\end{align}</math>

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.<ref>{{cite journal |year = 2015 |title = Archimedes on Spheres and Cylinders |journal = Math Pages |url = http://www.mathpages.com/home/kmath343/kmath343.htm}}</ref> [[File:Archimedes-spherical-cap.png|thumb|Archimedes' theorem that surface area of the region of sphere below horizontal plane H in given diagram is equal to area of a circle of radius t.]]In the above coloured diagram this radius is given as
<math display=block> 2r \sin \frac{\theta}{2}. </math>
In the adjacent black & white diagram this radius is given as "t".

Hence for a unit sphere the solid angle of the spherical cap is given as

<math display=block> \Omega = 4\pi \sin^2 \frac{\theta}{2} = 2\pi \left (1 - \cos\theta \right). </math>

{{anchor|hemisphere}}
When {{math|''θ''}} = {{sfrac|{{pi}}|2}}, the spherical cap becomes a [[Sphere|hemisphere]] having a solid angle 2{{pi}}.

The solid angle of the complement of the cone is
 
<math display=block>4\pi - \Omega = 2\pi \left(1 + \cos\theta \right) = 4\pi\cos^2 \frac{\theta}{2}.</math>

This is also the solid angle of the part of the [[celestial sphere]] that an astronomical observer positioned at latitude {{math|''θ''}} 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 {{mvar|''γ''}} from the cone's axis and passing through the cone's apex can be calculated by the formula<ref name = Mazonka>{{cite arXiv |last =  Mazonka |first = Oleg |year = 2012 |title = Solid Angle of Conical Surfaces, Polyhedral Cones, and Intersecting Spherical Caps |eprint=1205.1396 |class = math.MG}}</ref>

<math display=block> \Omega = 2 \left[ \arccos \left(\frac{\sin\gamma}{\sin\theta}\right) - \cos\theta \arccos\left(\frac{\tan\gamma}{\tan\theta}\right) \right]. </math>

For example, if {{math|1=''γ'' = −''θ''}}, then the formula reduces to the spherical cap formula above: the first term becomes {{pi}}, and the second {{math|{{pi}} cos ''θ''}}.

===Tetrahedron===
Let OABC be the vertices of a [[tetrahedron]] with an origin at O subtended by the triangular face ABC where <math>\vec a\ ,\, \vec b\ ,\, \vec c </math> are the vector positions of the vertices A, B and C. Define the [[vertex angle]] {{mvar|θ<sub>a</sub>}} to be the angle BOC and define {{mvar|θ<sub>b</sub>}}, {{mvar|θ<sub>c</sub>}} correspondingly. Let <math>\phi_{ab}</math> be the [[dihedral angle]] between the planes that contain the tetrahedral faces OAC and OBC and define <math>\phi_{ac}</math>, <math>\phi_{bc}</math> correspondingly. The solid angle {{math|Ω}} subtended by the triangular surface ABC is given by

<math display=block> \Omega = \left(\phi_{ab} + \phi_{bc} + \phi_{ac}\right)\ - \pi.</math>

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 {{pi}}"'', for the sum of the four internal solid angles of a tetrahedron as follows:

<math display=block> \sum_{i=1}^4 \Omega_i = 2 \sum_{i=1}^6 \phi_i\ - 4 \pi,</math>

where <math>\phi_i</math> ranges over all six of the dihedral angles between any two planes that contain the tetrahedral faces OAB, OAC, OBC and ABC.<ref>{{cite journal |last1=Hopf |first1=Heinz |title=Selected Chapters of Geometry |journal=ETH Zurich |date=1940 |pages=1–2 |url=http://pi.math.cornell.edu/~hatcher/Other/hopf-samelson.pdf |archive-url=https://web.archive.org/web/20180921122755/http://pi.math.cornell.edu/~hatcher/Other/hopf-samelson.pdf |archive-date=2018-09-21 |url-status=live}}</ref>

A useful formula for calculating the solid angle of the tetrahedron at the origin O that is purely a function of the vertex angles {{mvar|θ<sub>a</sub>}}, {{mvar|θ<sub>b</sub>}}, {{mvar|θ<sub>c</sub>}} is given by [[Simon Antoine Jean L'Huilier|L'Huilier]]'s theorem<ref>{{cite web|url=http://mathworld.wolfram.com/LHuiliersTheorem.html|title=L'Huilier's Theorem – from Wolfram MathWorld |publisher=Mathworld.wolfram.com |date=2015-10-19|access-date=2015-10-19}}</ref><ref>{{cite web|url=http://mathworld.wolfram.com/SphericalExcess.html|title=Spherical Excess – from Wolfram MathWorld |publisher=Mathworld.wolfram.com |date=2015-10-19|access-date=2015-10-19}}</ref> as

<math display=block> \tan \left( \frac{1}{4} \Omega \right) =
    \sqrt{ \tan \left( \frac{\theta_s}{2}\right) \tan \left( \frac{\theta_s - \theta_a}{2}\right) \tan \left( \frac{\theta_s - \theta_b}{2}\right) \tan \left(\frac{\theta_s - \theta_c}{2}\right)}, </math>

where
<math display=block> \theta_s = \frac {\theta_a + \theta_b + \theta_c}{2}. </math>

Another interesting formula involves expressing the vertices as vectors in 3 dimensional space. Let <math>\vec a\ ,\, \vec b\ ,\, \vec c </math> be the vector positions of the vertices A, B and C, and let {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} be the magnitude of each vector (the origin-point distance). The solid angle {{math|Ω}} subtended by the triangular surface ABC is:<ref>{{cite journal| first=Folke| last=Eriksson| title= On the measure of solid angles| journal= Math. Mag.| volume=63|issue=3|pages=184–187|year=1990| doi=10.2307/2691141| jstor=2691141}}</ref><ref>{{cite journal| last = Van Oosterom| first = A|author2=Strackee, J | year = 1983| title = The Solid Angle of a Plane Triangle| journal = IEEE Trans. Biomed. Eng.| volume = BME-30| issue = 2| pages = 125–126| doi = 10.1109/TBME.1983.325207| pmid = 6832789| s2cid = 22669644}}</ref>

<math display=block>\tan \left( \frac{1}{2} \Omega \right) =
  \frac{\left|\vec a\ \vec b\ \vec c\right|}{abc + \left(\vec a \cdot \vec b\right)c + \left(\vec a \cdot \vec c\right)b + \left(\vec b \cdot \vec c\right)a},
</math>

where
<math display=block>\left|\vec a\ \vec b\ \vec c\right|=\vec a \cdot (\vec b \times \vec c)</math>

denotes the [[triple product|scalar triple product]] of the three vectors and <math>\vec a \cdot \vec b</math> 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 {{mvar|a}}, {{mvar|b}}, {{mvar|c}} have the wrong [[determinant|winding]]. Computing the absolute value 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 {{pi}}.

===Pyramid===
The solid angle of a four-sided right rectangular [[Pyramid (geometry)|pyramid]] with [[apex (geometry)|apex]] angles {{mvar|a}} and {{mvar|b}} ([[dihedral angle]]s measured to the opposite side faces of the pyramid) is
<math display=block>\Omega = 4 \arcsin \left( \sin \left({a \over 2}\right) \sin \left({b \over 2}\right) \right). </math>

If both the side lengths ({{math|''α''}} and {{math|''β''}}) of the base of the pyramid and the distance ({{math|''d''}}) 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

<math display=block>\Omega = 4 \arctan \frac {\alpha\beta} {2d\sqrt{4d^2 + \alpha^2 + \beta^2}}. </math>

The solid angle of a right {{mvar|n}}-gonal pyramid, where the pyramid base is a regular {{mvar|n}}-sided polygon of circumradius {{mvar|r}}, with a 
pyramid height {{mvar|h}} is

<math display=block>\Omega = 2\pi - 2n \arctan\left(\frac {\tan \left({\pi\over n}\right)}{\sqrt{1 + {r^2 \over h^2}}} \right). </math>

The solid angle of an arbitrary pyramid with an {{math|''n''}}-sided base defined by the sequence of unit vectors representing edges {{math|{''s''<sub>1</sub>, ''s''<sub>2</sub>}, ... ''s''<sub>''n''</sub>}} can be efficiently computed by:<ref name ="Mazonka"/>

<math display=block> \Omega = 2\pi - \arg \prod_{j=1}^{n} \left(
    \left( s_{j-1} s_j \right)\left( s_{j} s_{j+1} \right) -
    \left( s_{j-1} s_{j+1} \right) +
    i\left[ s_{j-1} s_j s_{j+1} \right]
  \right).
</math>

where parentheses (* *) is a [[scalar product]] and square brackets [* * *] is a [[scalar triple product]], and {{mvar|i}} is an [[imaginary unit]]. Indices are cycled: {{math|''s''<sub>0</sub> {{=}} ''s''<sub>''n''</sub>}} and {{math|''s''<sub>1</sub> {{=}} ''s''<sub>''n'' + 1</sub>}}.  The complex products add the phase associated with each vertex angle of the polygon.  However, a multiple of
<math>2\pi</math> is lost in the branch cut of <math>\arg</math> and must be kept track of separately.  Also, the running product of complex phases must scaled occasionally to avoid underflow in the limit of nearly parallel segments.

=== Latitude-longitude rectangle ===
The solid angle of a latitude-longitude rectangle on a [[globe]] is
<math display=block>\left ( \sin \phi_\mathrm{N} - \sin \phi_\mathrm{S} \right ) \left ( \theta_\mathrm{E} - \theta_\mathrm{W} \,\! \right)\;\mathrm{sr},</math>
where {{math|''φ''<sub>N</sub>}} and {{math|''φ''<sub>S</sub>}} are north and south lines of [[latitude]] (measured from the [[equator]] in [[radian]]s with angle increasing northward), and {{math|''θ''<sub>E</sub>}} and {{math|''θ''<sub>W</sub>}} are east and west lines of [[longitude]] (where the angle in radians increases eastward).<ref>{{cite journal| year = 2003| title = Area of a Latitude-Longitude Rectangle| journal = The Math Forum @ Drexel| url = http://mathforum.org/library/drmath/view/63767.html}}</ref> Mathematically, this represents an arc of angle {{math|''ϕ''<sub>N</sub> − ''ϕ''<sub>S</sub>}} swept around a sphere by {{math|''θ''<sub>E</sub> − ''θ''<sub>W</sub>}} radians. When longitude spans 2{{pi}} radians and latitude spans {{pi}} 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.

=== Celestial objects ===
By using the definition of [[angular diameter]], the formula for the solid angle of a celestial object can be defined in terms of the radius of the object, <math display="inline">R</math>, and the distance from the observer to the object, <math>d</math>:

<math display=block>\Omega = 2 \pi \left (1 - \frac{\sqrt{d^2 - R^2}}{d} \right ) : d \geq R.</math>  

By inputting the appropriate average values for the [[Sun]] and the [[Moon]] (in relation to Earth), the average solid angle of the Sun is {{val|6.794|e=-5}} steradians and the average solid angle of the [[Moon]] is {{val|6.418|e=-5}} steradians. In terms of the total celestial sphere, the [[Sun]] and the [[Moon]] subtend average ''fractional areas'' of {{val|0.0005406}}% ({{val|5.406|u=[[part per million|ppm]]}}) and {{val|0.0005107}}% ({{val|5.107|u=ppm}}), respectively. As these solid angles are about the same size, the Moon can cause both total and annular solar [[Solar eclipse|eclipses]] depending on the distance between the Earth and the Moon during the eclipse.

== Solid angles in arbitrary dimensions ==
The solid angle subtended by the complete ({{mvar|d − 1}})-dimensional spherical surface of the unit sphere in [[Euclidean space|{{math|''d''}}-dimensional Euclidean space]] can be defined in any number of dimensions {{math|''d''}}. One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula
<math display="block">\Omega_{d} = \frac{2\pi^\frac{d}{2}}{\Gamma\left(\frac{d}{2}\right)}, </math>
where {{math|Γ}} is the [[gamma function]]. When {{math|''d''}} is an integer, the gamma function can be computed explicitly.<ref>{{cite journal| last = Jackson| first = FM| year = 1993| title = Polytopes in Euclidean n-space| journal = Bulletin of the Institute of Mathematics and Its Applications| volume = 29| issue = 11/12| pages = 172–174| url = https://www.researchgate.net/publication/265585180}}</ref> It follows that
<math display="block">
  \Omega_{d} = \begin{cases}
    \frac{1}{ \left(\frac{d}{2} - 1 \right)!} 2\pi^\frac{d}{2}\ & d\text{ even} \\
    \frac{\left(\frac{1}{2}\left(d - 1\right)\right)!}{(d - 1)!} 2^d \pi^{\frac{1}{2}(d - 1)}\ & d\text{ odd}.
  \end{cases}
</math>

This gives the expected results of 4{{pi}} steradians for the 3D sphere bounded by a surface of area {{math|4π''r''<sup>2</sup>}} and 2{{pi}} radians for the 2D circle bounded by a circumference of length {{math|2π''r''}}. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered  1D "sphere" is the interval {{closed-closed|−''r'', ''r''}} and this is bounded by two limiting points.

The counterpart to the vector formula in arbitrary dimension was derived by Aomoto<ref>{{cite journal|first=Kazuhiko| last=Aomoto| title=Analytic structure of Schläfli function| journal=Nagoya Math. J.| volume=68| year=1977| pages=1–16| doi=10.1017/s0027763000017839| doi-access=free}}</ref><ref>{{cite journal| last1=Beck|first1=M.|last2=Robins|first2=S.|last3=Sam|first3=S. V. |year=2010 |title=Positivity theorems for solid-angle polynomials |journal=Contributions to Algebra and Geometry |volume=51|issue=2| pages=493–507 |arxiv=0906.4031 |bibcode=2009arXiv0906.4031B}}</ref>
and independently by Ribando.<ref>{{cite journal| journal=Discrete & Computational Geometry| volume=36| issue=3| pages=479–487| year=2006| title= Measuring Solid Angles Beyond Dimension Three| first=Jason M.| last=Ribando| doi=10.1007/s00454-006-1253-4| doi-access=free}}</ref> It expresses them as an infinite multivariate [[Taylor series]]:
<math display="block">\Omega = \Omega_d \frac{\left|\det(V)\right|}{(4\pi)^{d/2}} \sum_{\vec a\in \N_0^{\binom {d}{2}}}
   \left [
     \frac{(-2)^{\sum_{i<j} a_{ij}}}{\prod_{i<j} a_{ij}!}\prod_i \Gamma \left (\frac{1+\sum_{m\neq i} a_{im}}{2} \right )
    \right ] \vec \alpha^{\vec a}.
 </math>
Given {{mvar|d}} unit vectors <math>\vec{v}_i</math> defining the angle, let {{mvar|V}} denote the matrix formed by combining them so the {{mvar|i}}th column is <math>\vec{v}_i</math>, and <math>\alpha_{ij} = \vec{v}_i\cdot\vec{v}_j = \alpha_{ji}, \alpha_{ii}=1</math>. The variables <math>\alpha_{ij},1 \le i < j \le d</math> form a multivariable <math>\vec \alpha = (\alpha_{12},\dotsc , \alpha_{1d}, \alpha_{23}, \dotsc, \alpha_{d-1,d}) \in \R^{\binom{d}{2}}</math>. For a "congruent" integer multiexponent <math>\vec a=(a_{12}, \dotsc, a_{1d}, a_{23}, \dotsc , a_{d-1,d}) \in \N_0^{\binom{d}{2}}, </math> define <math display="inline">\vec \alpha^{\vec a}=\prod \alpha_{ij}^{a_{ij}}</math>. Note that here <math>\N_0</math> = non-negative integers, or natural numbers beginning with 0. The notation <math>\alpha_{ji}</math> for <math>j > i</math> means the variable <math>\alpha_{ij}</math>, similarly for the exponents <math>a_{ji}</math>. 
Hence, the term <math display="inline">\sum_{m \ne l} a_{lm}</math> means the sum over all terms in <math>\vec a</math> 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.

== References ==
{{Reflist}}

== Further reading ==
*{{cite journal
|first1=A. H. |last1=Jaffey | title=Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables
|year=1954 | doi=10.1063/1.1771061 | journal=Rev. Sci. Instrum.
|pages=349–354 | volume=25
|issue=4 |bibcode=1954RScI...25..349J}}
*{{cite journal
|first1=A. Victor | last1=Masket |title=Solid angle contour integrals, series, and tables
|journal=Rev. Sci. Instrum. | volume=28 | number=3 | page=191 | year=1957 | doi =10.1063/1.1746479 | bibcode=1957RScI...28..191M
}}
*{{cite journal
|first1=Minoru | last1=Naito
|title=A method of calculating the solid angle subtended by a circular aperture
|journal=J. Phys. Soc. Jpn. | year=1957 | volume=12|number=10 | pages=1122–1129
|doi=10.1143/JPSJ.12.1122 |bibcode=1957JPSJ...12.1122N
}}
*{{cite journal
|first1=F. | last1=Paxton | title=Solid angle calculation for a circular disk
|volume=30 | number=4 | page=254 |year=1959
|journal= Rev. Sci. Instrum. | bibcode=1959RScI...30..254P
|doi=10.1063/1.1716590|doi-access=free }}
*{{cite journal
|first1=A. | last1=Khadjavi | title=Calculation of solid angle subtended by rectangular apertures
|volume=58 | number=10 | pages=1417–1418 |year=1968
|journal= J. Opt. Soc. Am.
|doi=10.1364/JOSA.58.001417
| bibcode=1968JOSA...58.1417K }}
*{{cite journal
|first1=R. P. | last1=Gardner | first2=A. | last2=Carnesale
|title=The solid angle subtended at a point by a circular disk
|volume=73 | number=2 | pages=228–230 | year=1969 | journal=Nucl. Instrum. Methods
|bibcode=1969NucIM..73..228G | doi=10.1016/0029-554X(69)90214-6
}}
*{{cite journal | first1=R. P. |last1=Gardner | first2=K. | last2=Verghese
|title=On the solid angle subtended by a circular disk | year=1971
|journal=Nucl. Instrum. Methods | volume=93 | number=1 | pages=163–167
|doi=10.1016/0029-554X(71)90155-8 | bibcode=1971NucIM..93..163G
}}
*{{cite journal | first1=H. |last1=Gotoh | first2=H. | last2=Yagi
|title=Solid angle subtended by a rectangular slit | year=1971
|journal=Nucl. Instrum. Methods | volume=96 | number=3 | pages=485–486
|doi=10.1016/0029-554X(71)90624-0
|bibcode=1971NucIM..96..485G }}
*{{cite journal | first1=J. |last1=Cook
|title=Solid angle subtended by a two rectangles | year=1980
|journal=Nucl. Instrum. Methods | volume=178 | number=2–3 | pages=561–564
|doi=10.1016/0029-554X(80)90838-1
|bibcode=1980NucIM.178..561C
}}
*{{cite journal
|first1=John S.. | last1=Asvestas | first2=David C. | last2=Englund
|title=Computing the solid angle subtended by a planar figure
|journal = Opt. Eng. | year=1994 | volume=33 | number=12 | pages=4055–4059 | doi= 10.1117/12.183402 | bibcode=1994OptEn..33.4055A
}} Erratum ibid. vol 50 (2011) page 059801.
*{{cite journal
|first1=Stanislaw | last1=Tryka | year=1997 | doi=10.1016/S0030-4018(96)00789-4
|title=Angular distribution of the solid angle at a point subtended by a circular disk
|journal=Opt. Commun. | volume=137 | number=4–6 | pages=317–333
|bibcode=1997OptCo.137..317T}}
*{{cite journal | first1=M. J. | last1=Prata | journal=Nucl. Instrum. Methods Phys. Res. A
|year=2004 | volume=521 | page=576| bibcode=2004NIMPA.521..576P 
|title=Analytical calculation of the solid angle subtended by a circular disc detector at a point cosine source
| issue=2–3 |doi=10.1016/j.nima.2003.10.098|arxiv=math-ph/0305034| s2cid=15266291 }}
*{{cite journal
|first1=D. M. | last1=Timus | first2= M. J. | last2=Prata | first3=S. L.
|last3=Kalla | first4=M. I. | last4=Abbas | first5=F. | last5=Oner
|first6=E. | last6=Galiano
|title=Some further analytical results on the solid angle subtended at a point by a circular disk using elliptic integrals
|journal=Nucl. Instrum. Methods Phys. Res. A | year=2007 | doi=10.1016/j.nima.2007.05.055
|volume=580 | pages=149–152
|bibcode=2007NIMPA.580..149T}}
{{commons category}}

== External links ==
*Arthur P. Norton, A Star Atlas, Gall and Inglis, Edinburgh, 1969.
*M. G. Kendall, A Course in the Geometry of N Dimensions, No. 8 of Griffin's Statistical Monographs & Courses, ed. M. G. Kendall, Charles Griffin & Co. Ltd, London, 1961
*{{mathworld|urlname=SolidAngle|title=Solid Angle}}

{{Classical mechanics derived SI units}}

{{DEFAULTSORT:Solid Angle}}
[[Category:Angle]]
[[Category:Euclidean solid geometry]]