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Solid angle

Article Discussion

Template:Short description Template:Distinguish Template:Infobox physical quantity In geometry, a solid angle (symbol: Template:Math) 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 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, <math>4\pi</math>. 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 (and therefore apparent size). This is evident during a solar eclipse.

Template:AnchorDefinition and propertiesEdit

Template:See also The magnitude of 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. 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.

File:Solid Angle, 1 Steradian.svg
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|Template:Pi]] sr. The solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2Template:Pi/3  sr. Template:AnchorThe solid angle subtended at the corner of a cube (an octant) or spanned by a spherical octant is Template:Pi/2  sr, one-eighth of the solid angle of a sphere.<ref name="u421">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Solid angles can also be measured in square degrees (1 sr = Template:Pars2 square degrees), in square arc-minutes and square arc-seconds, or in fractions of the sphere (1 sr = Template:Sfrac fractional area), also known as spat (1 sp = 4Template:Pi sr).

In spherical coordinates there is a formula for the differential,

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

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

The solid angle for an arbitrary oriented surface Template:Mvar subtended at a point Template:Mvar is equal to the solid angle of the projection of the surface Template:Mvar to the unit sphere with center Template:Mvar, 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 Template:Math with respect to point Template:Mvar, and where <math> \hat{n} </math> represents the unit normal vector to Template:Math. Even if the projection on the unit sphere to the surface Template:Mvar 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 Template:Math, orientation <math>\hat{n}</math>, and distance Template:Math 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 Template:Math.

Practical applicationsEdit

Solid angles for common objectsEdit

Cone, spherical cap, hemisphereEdit

File:Steradian cone and cap.svg
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 <math>r^2</math> only if solid angle of the cone is exactly 1 steradian. Hence, in this figure Template:Math and Template:Math.

The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2Template:Math, 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 Template:Math such that Template:Math this reduces to Template:Math, the area of a circle. (As Template:Math.)

The above is found by computing the following double integral using the unit 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>Template:Cite journal</ref>

File:Archimedes-spherical-cap.png
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>

Template:Anchor When Template:Math = Template:Sfrac, the spherical cap becomes a hemisphere having a solid angle 2Template: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 Template:Math 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 Template:Mvar from the cone's axis and passing through the cone's apex can be calculated by the formula<ref name = Mazonka>Template:Cite arXiv</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 Template:Math, then the formula reduces to the spherical cap formula above: the first term becomes Template:Pi, and the second Template:Math.

TetrahedronEdit

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 Template:Mvar to be the angle BOC and define Template:Mvar, Template:Mvar 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 Template:Math 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 Template: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>Template:Cite journal</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 Template:Mvar, Template:Mvar, Template:Mvar is given by L'Huilier's theorem<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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 Template:Mvar, Template:Mvar, and Template:Mvar be the magnitude of each vector (the origin-point distance). The solid angle Template:Math subtended by the triangular surface ABC is:<ref>Template:Cite journal</ref><ref>Template:Cite journal</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 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 Template:Mvar, Template:Mvar, Template:Mvar have the wrong 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 Template:Pi.

PyramidEdit

The solid angle of a four-sided right rectangular pyramid with apex angles Template:Mvar and Template:Mvar (dihedral angles 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 (Template:Math and Template:Math) of the base of the pyramid and the distance (Template:Math) 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 Template:Mvar-gonal pyramid, where the pyramid base is a regular Template:Mvar-sided polygon of circumradius Template:Mvar, with a pyramid height Template:Mvar 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 Template:Math-sided base defined by the sequence of unit vectors representing edges Template:Math 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 Template:Mvar is an imaginary unit. Indices are cycled: Template:Math and Template:Math. 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 rectangleEdit

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 Template:Math and Template:Math are north and south lines of latitude (measured from the equator in radians with angle increasing northward), and Template:Math and Template:Math are east and west lines of longitude (where the angle in radians increases eastward).<ref>Template:Cite journal</ref> Mathematically, this represents an arc of angle Template:Math swept around a sphere by Template:Math radians. When longitude spans 2Template:Pi radians and latitude spans Template: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 objectsEdit

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 Template:Val steradians and the average solid angle of the Moon is Template:Val steradians. In terms of the total celestial sphere, the Sun and the Moon subtend average fractional areas of Template:Val% (Template:Val) and Template:Val% (Template:Val), respectively. As these solid angles are about the same size, the Moon can cause both total and annular solar eclipses depending on the distance between the Earth and the Moon during the eclipse.

Solid angles in arbitrary dimensionsEdit

The solid angle subtended by the complete (Template:Mvar)-dimensional spherical surface of the unit sphere in [[Euclidean space|Template:Math-dimensional Euclidean space]] can be defined in any number of dimensions Template:Math. 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 Template:Math is the gamma function. When Template:Math is an integer, the gamma function can be computed explicitly.<ref>Template:Cite journal</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 4Template:Pi steradians for the 3D sphere bounded by a surface of area Template:Math and 2Template:Pi radians for the 2D circle bounded by a circumference of length Template:Math. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D "sphere" is the interval Template:Closed-closed and this is bounded by two limiting points.

The counterpart to the vector formula in arbitrary dimension was derived by Aomoto<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> and independently by Ribando.<ref>Template:Cite journal</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 Template:Mvar unit vectors <math>\vec{v}_i</math> defining the angle, let Template:Mvar denote the matrix formed by combining them so the Template:Mvarth 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.

ReferencesEdit

Template:Reflist

Further readingEdit

Template:Sister project

External linksEdit

  • 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
  • Template:Mathworld

Template:Classical mechanics derived SI units

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