Myujth

In geometry, a solid angle (symbol: 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 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, 4πdisplaystyle 4pi . 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:

Ω=Ar2displaystyle Omega =frac Ar^2

where A is the spherical surface area and r is the radius of the considered sphere.

Solid angles are often used in physics, 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.

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 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 degrees), in square minutes and square seconds, or in fractions of the sphere (1 sr = 1/4π fractional area), also known as spat (1 sp = 4π sr).

In spherical coordinates there is a formula for the differential,

dΩ=sin⁡θdθdφdisplaystyle dOmega =sin theta ,dtheta ,dvarphi

where θ is the colatitude (angle from the North pole) and φ 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:

Ω=∬Sr^⋅n^dΣr2 =∬Ssin⁡θdθdφdisplaystyle Omega =iint limits _Sfrac hat rcdot hat n,dSigma r^2 =iint limits _Ssin theta ,dtheta ,dvarphi

where r^=r→rdisplaystyle hat r=tfrac vec rr is the unit vector corresponding to r→displaystyle vec r, the position vector of an infinitesimal area of surface dΣ with respect to point P, and where n^displaystyle hat n represents the unit normal vector to dΣ. 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 r^⋅n^displaystyle hat rcdot hat n.

Thus one can approximate the solid angle subtended by a small facet having flat surface area dΣ, orientation n^displaystyle hat n, and distance r from the viewer as:

dΩ=4π(dΣA)(r^⋅n^)displaystyle dOmega =4pi left(frac dSigma Aright),(hat rcdot hat n)

where the surface area of a sphere is A = 4πr².

Practical applications

• Defining luminous intensity and luminance, and the correspondent radiometric quantities radiant intensity and radiance
  


• Calculating spherical excess E of a spherical triangle
  


• The calculation of potentials by using the boundary element method (BEM)


• Evaluating the size of ligands 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
  

Solid angles for common objects

Cone, spherical cap, hemisphere

Section of cone (1) and spherical cap (2) inside a sphere. In this figure θ = A/2 and r = 1.

The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2θ, is the area of a spherical cap on a unit sphere

Ω=2π(1−cos⁡θ) displaystyle Omega =2pi left(1-cos theta right)

For small θ such that cos θ ≈ 1 - θ²/2, this reduces to the area of a circle πθ².

The above is found by computing the following double integral using the unit surface element in spherical coordinates:

∫02π∫0θsin⁡θ′dθ′dϕ=2π∫0θsin⁡θ′dθ′ =2π[−cos⁡θ′]0θ =2π(1−cos⁡θ) displaystyle int _0^2pi int _0^theta sin theta ',dtheta ',dphi =2pi int _0^theta sin theta ',dtheta ' =2pi left[-cos theta 'right]_0^theta =2pi left(1-cos theta right)

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.^[1] In the diagram this radius is given as:

2rsin⁡(θ2)displaystyle 2rsin left(frac theta 2right)

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

Ω=4πsin2⁡(θ2)=2π(1−cos⁡θ) displaystyle Omega =4pi sin ^2left(frac theta 2right)=2pi left(1-cos theta right)

When θ = π/2, the spherical cap becomes a hemisphere having a solid angle 2π.

The solid angle of the complement of the cone (picture a melon with the cone cut out) is clearly:

4π−Ω=2π(1+cos⁡θ)displaystyle 4pi -Omega =2pi left(1+cos theta right)

This is also the solid angle of the part of the celestial sphere that a Terran astronomical observer positioned at latitude θ can see as the earth rotates. At the equator you see all of the celestial sphere, at either pole only one half.

The solid angle subtended by a segment of a spherical cap cut by a plane at angle γ from the cone's axis and passing through the cone's apex can be calculated by the formula:^[2]

Ω=2[arccos⁡(sin⁡γsin⁡θ)−cos⁡θarccos⁡(tan⁡γtan⁡θ)] displaystyle Omega =2left[arccos left(frac sin gamma sin theta right)-cos theta arccos left(frac tan gamma tan theta right)right]

For example, if γ=-θ, then the formula reduces to the spherical cap formula above: the first term becomes π, and the second πcosθ.

Tetrahedron

Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where a→ ,b→ ,c→displaystyle vec a ,,vec b ,,vec c are the vector positions of the vertices A, B and C. Define the vertex angle θ_a to be the angle BOC and define θ_b, θ_c correspondingly. Let φ_ab be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define φ_ac, φ_bc correspondingly. The solid angle Ω subtended by the triangular surface ABC is given by

Ω=(ϕab+ϕbc+ϕac) −π displaystyle Omega =left(phi _ab+phi _bc+phi _acright) -pi

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 π", for the sum of the four internal solid angles of a tetrahedron as follows:

∑i=14Ωi=2∑i=16ϕi −4π displaystyle sum _i=1^4Omega _i=2sum _i=1^6phi _i -4pi

where φ_i 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 Ω subtended by the triangular surface ABC where a→ ,b→ ,c→displaystyle vec a ,,vec b ,,vec c are the vector positions of the vertices A, B and C has been given by Oosterom and Strackee ^[3] (although the result was known earlier by Euler and Lagrange^[4]):

tan⁡(12Ω)=|a→ b→ c→|abc+(a→⋅b→)c+(a→⋅c→)b+(b→⋅c→)adisplaystyle tan left(frac 12Omega right)=frac abc+left(vec acdot vec bright)c+left(vec acdot vec cright)b+left(vec bcdot vec cright)a

where

|a→ b→ c→|=a→⋅(b→×c→)=vec acdot (vec btimes vec c)

denotes the scalar triple product of the three vectors;

a→displaystyle vec a is the vector representation of point A, while a is the magnitude of that vector (the origin-point distance)

a→⋅b→displaystyle vec acdot vec b denotes the scalar product.

When implementing the above equation care must be taken with the atan function to avoid negative or incorrect solid angles. One source of potential errors is that the scalar triple product can be negative if a, b, c have the wrong winding. Computing abs(det) 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 atan returns a negative value that must be increased by π.

Another useful formula for calculating the solid angle of the tetrahedron at the origin O that is purely a function of the vertex angles θ_a, θ_b, θ_c is given by L'Huilier's theorem^[5][6] as

tan⁡(14Ω)=tan⁡(θs2)tan⁡(θs−θa2)tan⁡(θs−θb2)tan⁡(θs−θc2)displaystyle tan left(frac 14Omega right)=sqrt tan left(frac theta _s2right)tan left(frac theta _s-theta _a2right)tan left(frac theta _s-theta _b2right)tan left(frac theta _s-theta _c2right)

where

θs=θa+θb+θc2displaystyle theta _s=frac theta _a+theta _b+theta _c2

Pyramid

The solid angle of a four-sided right rectangular pyramid with apex angles a and b (dihedral angles measured to the opposite side faces of the pyramid) is

Ω=4arcsin⁡(sin⁡(a2)sin⁡(b2))displaystyle Omega =4arcsin left(sin left(a over 2right)sin left(b over 2right)right)

If both the side lengths (α and β) of the base of the pyramid and the distance (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

Ω=4arctan⁡αβ2d4d2+α2+β2displaystyle Omega =4arctan frac alpha beta 2dsqrt 4d^2+alpha ^2+beta ^2

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

Ω=2π−2narctan⁡(tan⁡(πn)1+r2h2)displaystyle Omega =2pi -2narctan left(frac tan left(pi over nright)sqrt 1+r^2 over h^2right)

The solid angle of an arbitrary pyramid with an n-sided base defined by the sequence of unit vectors representing edges s₁, s₂, ... s_n can be efficiently computed by:^[2]

Ω=2π−arg⁡∏j=1n((sj−1sj)(sjsj+1)−(sj−1sj+1)+i[sj−1sjsj+1])displaystyle Omega =2pi -arg prod _j=1^nleft(left(s_j-1s_jright)left(s_js_j+1right)-left(s_j-1s_j+1right)+ileft[s_j-1s_js_j+1right]right)

where parentheses (* *) is a scalar product and square brackets [* * *] is a scalar triple product, and i is an imaginary unit. Indices are cycled: s₀ = s_n and s₁ = s_{n + 1}.

Latitude-longitude rectangle

The solid angle of a latitude-longitude rectangle on a globe is

(sin⁡ϕN−sin⁡ϕS)(θE−θW)srdisplaystyle left(sin phi _mathrm N -sin phi _mathrm S right)left(theta _mathrm E -theta _mathrm W ,!right);mathrm sr ,

where φ_N and φ_S are north and south lines of latitude (measured from the equator in radians with angle increasing northward), and θ_E and θ_W are east and west lines of longitude (where the angle in radians increases eastward).^[7] Mathematically, this represents an arc of angle φ_N − φ_S swept around a sphere by θ_E − θ_W radians. When longitude spans 2π radians and latitude spans π 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 9.310×10⁻³ radians. The Moon is seen from Earth at an average diameter of 9.22×10⁻³ radians. We can substitute these into the equation given above for the solid angle subtended by a cone with apex angle 2θ:

Ω=2π(1−cos⁡θ)displaystyle Omega =2pi left(1-cos theta right)

The resulting value for the Sun is 6.807×10⁻⁵ steradians. The resulting value for the Moon is 6.67×10⁻⁵ steradians. In terms of the total celestial sphere, the Sun and the Moon subtend fractional areas of 0.000542% (Sun) and 0.000531% (Moon). 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 (d − 1)-dimensional spherical surface of the unit sphere in d-dimensional Euclidean space can be defined in any number of dimensions d. One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula

Ωd=2πd2Γ(d2) displaystyle Omega _d=frac 2pi ^frac d2Gamma left(frac d2right)

where Γ is the gamma function. When d is an integer, the gamma function can be computed explicitly.^[8] It follows that

Ωd={1(d2−1)!2πd2 d even(12(d−1))!(d−1)!2dπ12(d−1) d odddisplaystyle Omega _d=begincasesfrac 1left(frac d2-1right)!2pi ^frac d2 &dtext even\frac left(frac 12left(d-1right)right)!(d-1)!2^dpi ^frac 12(d-1) &dtext oddendcases

This gives the expected results of 4π steradians for the 3D sphere bounded by a surface of area 4πr² and 2π radians for the 2D circle bounded by a circumference of length 2πr. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D "sphere" is the interval [ −r, r ] and this is bounded by two limiting points.

The counterpart to the vector formula in arbitrary dimension was derived by Aomoto
^[9][10]
and independently by Ribando.^[11] It expresses them as an infinite multivariate Taylor series:

Ω=Ωd|det(V)|(4π)d/2∑a∈N(d2)[(−2)∑i<jaij∏i<jaij!∏iΓ(1+∑m≠iaim2)]αadisplaystyle Omega =Omega _dfrac (4pi )^d/2sum _ain mathbb N ^binom d2left[frac (-2)^sum _i<ja_ijprod _i<ja_ij!prod _iGamma left(frac 1+sum _mneq ia_im2right)right]alpha ^a

Given d unit vectors vi→displaystyle vec v_i defining the angle, V denotes the matrix formed by combining them so the i:th column is vi→displaystyle vec v_i, and αij=vi→⋅vj→displaystyle alpha _ij=vec v_icdot vec v_j. Where this series converges, it converges to the solid angle defined by the vectors.

See also

• Transcendent angle


• Versine and vercosine

References

Further reading

• 
  Jaffey, A. H. (1954). "Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables". Rev. Sci. Instrum. 25. pp. 349–354. Bibcode:1954RScI...25..349J. doi:10.1063/1.1771061. 
  


• 
  Masket, A. Victor (1957). "Solid angle contour integrals, series, and tables". Rev. Sci. Instrum. 28 (3). p. 191. Bibcode:1957RScI...28..191M. doi:10.1063/1.1746479. 
  


• 
  Naito, Minoru (1957). "A method of calculating the solid angle subtended by a circular aperture". J. Phys. Soc. Jpn. 12 (10). pp. 1122–1129. Bibcode:1957JPSJ...12.1122N. doi:10.1143/JPSJ.12.1122. 
  


• 
  Paxton, F. (1959). "Solid angle calculation for a circular disk". Rev. Sci. Instrum. 30 (4). p. 254. Bibcode:1959RScI...30..254P. doi:10.1063/1.1716590. 
  


• 
  Gardner, R. P.; Carnesale, A. (1969). "The solid angle subtended at a point by a circular disk". Nucl. Instrum. Methods. 73 (2). pp. 228–230. Bibcode:1969NucIM..73..228G. doi:10.1016/0029-554X(69)90214-6. 
  


• 
  Gardner, R. P.; Verghese, K. (1971). "On the solid angle subtended by a circular disk". Nucl. Instrum. Methods. 93 (1). pp. 163–167. Bibcode:1971NucIM..93..163G. doi:10.1016/0029-554X(71)90155-8. 
  


• 
  Asvestas, John S..; Englund, David C. (1994). "Computing the solid angle subtended by a planar figure". Opt. Eng. 33 (12). pp. 4055–4059. Bibcode:1994OptEn..33.4055A. doi:10.1117/12.183402. 
  


• 
  Tryka, Stanislaw (1997). "Angular distribution of the solid angle at a point subtended by a circular disk". Opt. Commun. 137 (4-6). pp. 317–333. Bibcode:1997OptCo.137..317T. doi:10.1016/S0030-4018(96)00789-4. 
  


• 
  Prata, M. J. (2004). "Analytical calculation of the solid angle subtended by a circular disc detector at a point cosine source". Nucl. Instrum. Methods Phys. Res. A. 521. p. 576. arXiv:math-ph/0305034 . Bibcode:2004NIMPA.521..576P. doi:10.1016/j.nima.2003.10.098. 
  


• 
  Timus, D. M.; Prata, M. J.; Kalla, S. L.; Abbas, M. I.; Oner, F.; Galiano, E. (2007). "Some further analytical results on the solid angle subtended at a point by a circular disk using elliptic integrals". Nucl. Instrum. Methods Phys. Res. A. 580. pp. 149–152. Bibcode:2007NIMPA.580..149T. doi:10.1016/j.nima.2007.05.055. 
  

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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


• Weisstein, Eric W. "Solid Angle". MathWorld.