Solid angle

 ( 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 Subtended angle its solid angle at that point.

In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr), which is equal to one square radian, sr = rad². 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, $4\backslash pi$. Solid angles can also be measured in squares of angular measures such as Square degree, 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.

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Definition and properties The magnitude of an object's solid angle in is equal to the 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

$$\backslash Omega=\backslash frac\{A\}\{r^2\},$$

where $A$ is the area (of any shape) on the surface of the sphere and $r$ 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.

The solid angle of a sphere measured from any point in its interior is 4 sr. The solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2/3  sr. The solid angle subtended at the corner of a cube (an octant) or spanned by a spherical octant is /2  sr, one-eighth of the solid angle of a sphere.

Solid angles can also be measured in (1 sr = ² square degrees), in square arc-minutes and square arc-seconds. It can also be expressed 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,

$$d\backslash Omega\; =\; \backslash sin\backslash theta\backslash ,d\backslash theta\backslash ,d\backslash varphi,$$

where is the colatitude (angle from the North Pole) and is the longitude.

The solid angle for an arbitrary oriented surface subtended at a point is equal to the solid angle of the projection of the surface to the unit sphere with center , which can be calculated as the surface integral:

$$\backslash Omega\; =\; \backslash iint\_S\; \backslash frac\{\; \backslash hat\{r\}\; \backslash cdot\; \backslash hat\{n\}\}\{r^2\}\backslash ,dS\; \backslash \; =\; \backslash iint\_S\; \backslash sin\backslash theta\backslash ,d\backslash theta\backslash ,d\backslash varphi,$$

where $\backslash hat\{r\}\; =\; \backslash vec\{r\}\; /\; r$ is the unit vector corresponding to $\backslash vec\{r\}$, the position vector of an infinitesimal area of surface with respect to point , and where $\backslash hat\{n\}$ represents the unit normal vector to . Even if the projection on the unit sphere to the surface is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product $\backslash hat\{r\}\; \backslash cdot\; \backslash hat\{n\}$.

Thus one can approximate the solid angle subtended by a small facet having flat surface area , orientation $\backslash hat\{n\}$, and distance from the viewer as:

$$d\backslash Omega\; =\; 4\; \backslash pi\; \backslash left(\backslash frac\{dS\}\{A\}\backslash right)\; \backslash ,\; (\backslash hat\{r\}\; \backslash cdot\; \backslash hat\{n\}),$$

where the surface area of a sphere is .

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Practical applications

• Defining luminous intensity and luminance, and the correspondent radiometric quantities radiant intensity and radiance
• Calculating spherical excess of a spherical triangle
• The calculation of potentials by using the boundary element method (BEM)
• Evaluating the size of 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.

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Solid angles for common objects

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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

$$\backslash Omega\; =\; 2\backslash pi\; \backslash left\; (1\; -\; \backslash cos\backslash theta\; \backslash right)\backslash \; =\; 4\backslash pi\; \backslash sin^2\; \backslash frac\{\backslash theta\}\{2\}.$$

For small , such that , this reduces to = .

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

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. the above coloured diagram this radius is given as $$2r\; \backslash sin\; \backslash frac\{\backslash theta\}\{2\}.$$ 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

$$\backslash Omega\; =\; 4\backslash pi\; \backslash sin^2\; \backslash frac\{\backslash theta\}\{2\}\; =\; 2\backslash pi\; \backslash left\; (1\; -\; \backslash cos\backslash theta\; \backslash right).$$

When = , the spherical cap becomes a Sphere having a solid angle 2.

The solid angle of the complement of the cone is

$$4\backslash pi\; -\; \backslash Omega\; =\; 2\backslash pi\; \backslash left(1\; +\; \backslash cos\backslash theta\; \backslash right)\; =\; 4\backslash pi\backslash cos^2\; \backslash frac\{\backslash theta\}\{2\}.$$

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

$$\backslash Omega\; =\; 2\; \backslash left.$$

For example, if , then the formula reduces to the spherical cap formula above: the first term becomes , and the second .

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Tetrahedron Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where $\backslash vec\; a\backslash \; ,\backslash ,\; \backslash vec\; b\backslash \; ,\backslash ,\; \backslash vec\; c$ are the vector positions of the vertices A, B and C. Define the vertex angle to be the angle BOC and define , correspondingly. Let $\backslash phi\_\{ab\}$ be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define $\backslash phi\_\{ac\}$, $\backslash phi\_\{bc\}$ correspondingly. The solid angle subtended by the triangular surface ABC is given by

$$\backslash Omega\; =\; \backslash left(\backslash phi\_\{ab\}\; +\; \backslash phi\_\{bc\}\; +\; \backslash phi\_\{ac\}\backslash right)\backslash \; -\; \backslash pi.$$

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

$$\backslash sum\_\{i=1\}^4\; \backslash Omega\_i\; =\; 2\; \backslash sum\_\{i=1\}^6\; \backslash phi\_i\backslash \; -\; 4\; \backslash pi,$$

where $\backslash phi\_i$ 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 of the tetrahedron at the origin O that is purely a function of the vertex angles , , is given by L'Huilier's theorem as

$$\backslash tan\; \backslash left(\; \backslash frac\{1\}\{4\}\; \backslash Omega\; \backslash right)\; =$$

   \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)}, 
     

where $$\backslash theta\_s\; =\; \backslash frac\; \{\backslash theta\_a\; +\; \backslash theta\_b\; +\; \backslash theta\_c\}\{2\}.$$

Another interesting formula involves expressing the vertices as vectors in 3 dimensional space. Let $\backslash vec\; a\backslash \; ,\backslash ,\; \backslash vec\; b\backslash \; ,\backslash ,\; \backslash vec\; c$ 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:

$$\backslash tan\; \backslash left(\; \backslash frac\{1\}\{2\}\; \backslash Omega\; \backslash right)\; =$$

 \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},
     

where $$\backslash left|\backslash vec\; a\backslash \; \backslash vec\; b\backslash \; \backslash vec\; c\backslash right|=\backslash vec\; a\; \backslash cdot\; (\backslash vec\; b\; \backslash times\; \backslash vec\; c)$$

denotes the triple product of the three vectors and $\backslash vec\; a\; \backslash cdot\; \backslash vec\; b$ 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 determinant. 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 .

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Pyramid The solid angle of a four-sided right rectangular pyramid with apex angles and ( measured to the opposite side faces of the pyramid) is $$\backslash Omega\; =\; 4\; \backslash arcsin\; \backslash left(\; \backslash sin\; \backslash left(\{a\; \backslash over\; 2\}\backslash right)\; \backslash sin\; \backslash left(\{b\; \backslash over\; 2\}\backslash right)\; \backslash right).$$

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

$$\backslash Omega\; =\; 4\; \backslash arctan\; \backslash frac\; \{\backslash alpha\backslash beta\}\; \{2d\backslash sqrt\{4d^2\; +\; \backslash alpha^2\; +\; \backslash beta^2\}\}.$$

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

$$\backslash Omega\; =\; 2\backslash pi\; -\; 2n\; \backslash arctan\backslash left(\backslash frac\; \{\backslash tan\; \backslash left(\{\backslash pi\backslash over\; n\}\backslash right)\}\{\backslash sqrt\{1\; +\; \{r^2\; \backslash over\; h^2\}\}\}\; \backslash right).$$

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:

$$\backslash Omega\; =\; 2\backslash pi\; -\; \backslash arg\; \backslash prod\_\{j=1\}^\{n\}\; \backslash left($$

   \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).
     

where parentheses (* *) is a scalar product and square brackets * is a scalar triple product, and is an imaginary unit. Indices are cycled: and . The complex products add the phase associated with each vertex angle of the polygon. However, a multiple of $2\backslash pi$ is lost in the branch cut of $\backslash arg$ and must be kept track of separately. Also, the running product of complex phases must be scaled occasionally to avoid underflow in the limit of nearly parallel segments.

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Latitude-longitude rectangle The solid angle of a latitude-longitude rectangle on a globe is $$\backslash left\; (\; \backslash sin\; \backslash phi\_\backslash mathrm\{N\}\; -\; \backslash sin\; \backslash phi\_\backslash mathrm\{S\}\; \backslash right\; )\; \backslash left\; (\; \backslash theta\_\backslash mathrm\{E\}\; -\; \backslash theta\_\backslash mathrm\{W\}\; \backslash ,\backslash !\; \backslash right)\backslash ;\backslash mathrm\{sr\},$$ where and are north and south lines of latitude (measured from the equator in 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.

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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, $R$, and the distance from the observer to the object, $d$:

$$\backslash Omega\; =\; 2\; \backslash pi\; \backslash left\; (1\; -\; \backslash frac\{\backslash sqrt\{d^2\; -\; R^2\}\}\{d\}\; \backslash right\; )\; :\; d\; \backslash geq\; R.$$

By inputting the appropriate average values for the Sun and the Moon (in relation to Earth), the average solid angle of the Sun is steradians and the average solid angle of the Moon is steradians. In terms of the total celestial sphere, the Sun and the Moon subtend average fractional areas of % () and % (), respectively. As these solid angles are about the same size, the Moon can cause both total and annular solar Solar eclipse depending on the distance between the Earth and the Moon during the eclipse.

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Solid angles in arbitrary dimensions The solid angle subtended by the complete ()-dimensional spherical surface of the unit sphere in 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 $$\backslash Omega\_\{d\}\; =\; \backslash frac\{2\backslash pi^\backslash frac\{d\}\{2\}\}\{\backslash Gamma\backslash left(\backslash frac\{d\}\{2\}\backslash right)\},$$ where is the gamma function. When is an integer, the gamma function can be computed explicitly. It follows that $$$$

 \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}
     

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:

  \left [
    \frac{(-2)^{\sum_{i
     

Given unit vectors $\backslash vec\{v\}\_i$ defining the angle, let denote the matrix formed by combining them so the th column is $\backslash vec\{v\}\_i$, and $\backslash alpha\_\{ij\}\; =\; \backslash vec\{v\}\_i\backslash cdot\backslash vec\{v\}\_j\; =\; \backslash alpha\_\{ji\},\; \backslash alpha\_\{ii\}=1$. The variables form a multivariable $\backslash vec\; \backslash alpha\; =\; (\backslash alpha\_\{12\},\backslash dotsc\; ,\; \backslash alpha\_\{1d\},\; \backslash alpha\_\{23\},\; \backslash dotsc,\; \backslash alpha\_\{d-1,d\})\; \backslash in\; \backslash R^\{\backslash binom\{d\}\{2\}\}$. For a "congruent" integer multiexponent $\backslash vec\; a=(a\_\{12\},\; \backslash dotsc,\; a\_\{1d\},\; a\_\{23\},\; \backslash dotsc\; ,\; a\_\{d-1,d\})\; \backslash in\; \backslash N\_0^\{\backslash binom\{d\}\{2\}\},$ define $\backslash vec\; \backslash alpha^\{\backslash vec\; a\}=\backslash prod\; \backslash alpha\_\{ij\}^\{a\_\{ij\}\}$. Note that here $\backslash N\_0$ = non-negative integers, or natural numbers beginning with 0. The notation $\backslash alpha\_\{ji\}$ for $j\; >\; i$ means the variable $\backslash alpha\_\{ij\}$, similarly for the exponents $a\_\{ji\}$. Hence, the term $\backslash sum\_\{m\; \backslash ne\; l\}\; a\_\{lm\}$ means the sum over all terms in $\backslash vec\; a$ 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.

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Notes

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Further reading

• Erratum ibid. vol 50 (2011) page 059801.

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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

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