NationMaster - Encyclopedia: Solid angle

The solid angle, Ω, is the angle in three-dimensional space that an object subtends at a point. It is a measure of how big that object appears to an observer looking from that point. For instance, a small object nearby could subtend the same solid angle as a large object far away. The solid angle is proportional to the surface area, S, of a projection of that object onto a sphere centered at that point, divided by the square of the sphere's radius, R. (Symbolically, Ω = k S/R², where k is the proportionality constant.) A solid angle is related to the surface of a sphere in the same way an ordinary angle is related to the circumference of a circle. In mathematics, subtended usually refers to the direct relationship between an angle and its arc length, or for solid angle the area on a unit sphere cut out by the envelope of the vectors defining the perimeter. ... Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. ... For other uses, see Sphere (disambiguation). ... This article is about angles in geometry. ... The circumference is the distance around a closed curve. ... This article is about the shape and mathematical concept of circle. ...

If the proportionality constant is chosen to be 1, the units of solid angle will be the SI steradian (abbreviated "sr"). Thus the solid angle of a sphere measured from a point in its interior is 4π sr, and the solid angle subtended at the center of a cube by one of its sides is one-sixth of that, or 2π/3 sr. Solid angles can also be measured (for k = (180/π)²) in square degrees or (for k = 1/4π) in fractions of the sphere (i.e., fractional area). Look up si, Si, SI in Wiktionary, the free dictionary. ... A graphical representation of 1 steradian. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... A square degree is a non-SI unit that can be used to measure solid angles (that is, the area of the projection of a surface onto a unit sphere centered on the point of observation). ...

One way to determine the fractional area subtended by a spherical surface is to divide the area of that surface by the entire surface area of the sphere. The fractional area can then be converted to steradian or square degree measurements by the following formulae:

To obtain the solid angle in steradians, multiply the fractional area by 4π.
To obtain the solid angle in square degrees, multiply the fractional area by 4π × (180/π)², which is equal to 129600/π.

More rigorously, the solid angle for a surface S subtended at a point P is given by the surface integral:

$Omega = iint_S frac { vec{r} cdot vec{n} dS }{r^3}.$

where $vec{r}$ is the vector position of an infinitesimal area of surface $, dS$ with respect to point P and where $vec n$ represents the unit vector normal to $, dS$ . This article is about vectors that have a particular relation to the spatial coordinates. ...

1 Practical applications
2 Solid angles for common objects
3 Solid angles in arbitrary dimensions
4 References

Practical applications

Defining luminous intensity and luminance
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.

Luminous intensity is a measure of the energy emitted by a light source in a particular direction. ... Luminance (also called luminosity) is a photometric measure of the density of luminous intensity in a given direction. ... Right spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i. ... In chemistry, a ligand is an atom, ion, or molecule (see also: functional group) that generally donates one or more of its electrons through a coordinate covalent bond to, or shares its electrons through a covalent bond with, one or more central atoms or ions (these ligands act as a... Ligand cone angle is a measure of the size of a ligand. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... For the indie-pop band, see The Magnetic Fields. ...

Solid angles for common objects

Tetrahedron

Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where $vec a ,, vec b ,, vec c$ are the vector positions of the vertices A, B and C. Define the vertex angle $theta_a ,$ to be the angle BOC and define $theta_b ,, theta_c$ correspondingly. Let $phi_{ab} ,$ be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define $phi_{bc} ,, phi_{ac}$ correspondingly. The solid angle at O subtended by the triangular surface ABC is given by For the academic journal, see Tetrahedron (journal). ... In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ...

$Omega = phi_{ab} + phi_{bc} + phi_{ac} - pi. ,$

This follows from the theory of spherical excess and it leads on to the fact that there is an analogous theorem to the sum of internal angles of a triangle equal to $π$ , for the sum of the four internal solid angles of a tetrahedron as follows: Right spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ...

$sum_{i=1}^4 Omega_i = 2 sum_{i=1}^6 phi_i - 4 pi$

where $phi_i ,$ ranges over all six of the dihedral angles between any two planes that contain the tetrahedral faces OAB, OAC, OBC and ABC.

An efficient algorithm for calculating the solid angle at O that subtends the triangular surface ABC where $vec a ,, vec b ,, vec c$ are the vector positions of the vertices A, B and C has been given by Oosterom and Strackee (IEEE Trans. Biom. Eng., Vol BME-30, No 2, 1983):

$tan left( frac{1}{2} Omega right) = frac{[vec a vec b vec c]}{ abc + (vec a cdot vec b)c + (vec a cdot vec c)b + (vec b cdot vec c)a},$

where

$[vec a vec b vec c]$

denotes the determinant of the matrix that results when writing the vectors together in a row, e.g. $M_{i1}=vec a_i$ and so on--this is also equivalent to the scalar triple product of the three vectors; In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In vector calculus, there are two ways of multiplying three vectors together, to make a triple product of vectors. ...

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

$vec a cdot vec b$ denotes the scalar product.

Another useful formula for calculating the solid angle of the tetrahedron at the origin O that is purely a function of the vertex angles $theta_a ,, theta_b ,, theta_c$ is given by L' Huilier's theorem as In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ...

$tan left( frac{1}{4} Omega right) = sqrt{ tan left( frac{theta_s}{2}right) tan left( frac{theta_s - theta_a}{2}right) tan left( frac{theta_s - theta_b}{2}right) tan left( frac{theta_s - theta_c}{2}right)}$

where

$theta_s = frac {theta_a + theta_b + theta_c}{2}.$

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 apex angle $2 theta ,!$ , is the area of a spherical cap on a unit sphere This article is about the geometric object, for other uses see Cone. ... In geometry, an apex is a descriptive label for a visual singular highest or most distant point or vertex in an isosceles triangle, pyramid or cone, usually contrasting with the opposite side called the base. ... In geometry, a spherical cap is a portion of a sphere cut off by a plane. ... In mathematics, unit ball and unit sphere refer to a ball with radius equal to 1. ...

$Omega = 2 pi left (1 - cos {theta} right) .,!$

(The above result is found by computing the following double integral using the unit surface element in spherical polars): In mathematical analysis, there is a serious distinction between a double integral and an iterated integral. ...

$int_0^{2pi} int_0^{theta} sin theta' , d theta' , d phi = 2piint_0^{theta} sin theta' , d theta' = 2pileft[ -cos theta' right]_0^{theta} = 2pileft(1 -cos theta right).$

Over 2200 years ago Archimedes proved, without the use of calculus, that the surface area of a spherical cap mapped identically onto the area of a circle whose radius was equal to the distance from the rim of the spherical cap to the lowest point on the surface of the spherical cap below the rim. In the diagram opposite this radius is given as: For other uses, see Archimedes (disambiguation). ... For other uses, see Calculus (disambiguation). ...

$2r sin left( frac{ theta}{2} right).,$

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

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

When θ = π/2, the spherical cap becomes a hemisphere having a solid angle 2π. For other uses, see Sphere (disambiguation). ...

Pyramid

The solid angle of a four-sided right rectangular pyramid with apex angles For other meanings, see pyramid (disambiguation). ...

$a ,!$ and $b ,!$ (dihedral angles measured to the opposite side faces of the pyramid) is $4 arcsin left (sin {a over 2} sin {b over 2} right). ,!$

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 sfere) are known, then the above equation can be manipulated to give In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ... In geometry, an apex is a descriptive label for a visual singular highest or most distant point or vertex in an isosceles triangle, pyramid or cone, usually contrasting with the opposite side called the base. ...

$Omega = 4 arcsin frac {alphabeta} {sqrt{(4d^2+alpha^2)(4d^2+beta^2)}}. ,$

Latitude-longitude rectangle

The solid angle of a latitude-longitude rectangle on a globe is $left ( sin phi_N - sin phi_S right ) left ( theta_E - theta_W ,! right)$ , where $phi_N ,!$ and $phi_S ,!$ are north and south lines of latitude (measured from the equator in radians with angle increasing northward), and $theta_E ,!$ and $theta_W ,!$ are east and west lines of longitude (where the angle in radians increases eastward).[1] Mathematically, this represents an arc of angle $phi_N - phi_S ,!$ swept around a sphere by $theta_E - theta_W ,!$ radians. When longitude spans 2π radians and latitude spans π radians, the solid angle is that of a sphere. This article is about a spherical model of the Earth, or similar. ... This article is about the geographical term. ... World map showing the equator in red For other uses, see Equator (disambiguation). ... In mathematics and physics, the radian is a unit of angle measure. ... Longitude is the east-west geographic coordinate measurement most commonly utilized in cartography and global navigation. ...

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. For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the...

Sun and Moon

The Sun and Moon are both seen from Earth at a fractional area of 0.001% of the celestial hemisphere or about 6×10^-5 steradian.[2] Sol redirects here. ... This article is about Earths moon. ...

Solid angles in arbitrary dimensions

The solid angle subtended by the full surface of the unit n-sphere 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 For other uses, see sphere (disambiguation). ...

$Omega_{d} = frac{2pi^{d/2}}{Gamma left (frac{d}{2} right )} qquad forall d in mathbb{N}$

where $Γ$ is the Gamma function. Since $d$ is an integer, the Gamma function can be computed explicitly. It follows that The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Î“) is an extension of the factorial function to real and complex numbers. ...

$Omega_{d} = begin{cases} frac{dpi^{d/2}}{ left (frac{d}{2} right )!} & d = dot{2} frac{2^dleft (frac{d-1}{2} right ) !}{(d-1)!} pi^{(d-1)/2} & d ne dot{2} end{cases}$

This gives the expected results of 2π rad for the 2D circumference and 4π srad for the 3D sphere. It also throws the slightly less obvious 2 for the 1D case, in which the origin-centered unit "sphere" is the interval $[ - 1,1]$ , which indeed has a measure of 2. A graphical representation of 1 steradian. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ...

References

V. C. A. Ferraro, Electromagnetic Theory, Athlone Press, University of London, pp 42-44, 1962
Arthur P. Norton, A Star Atlas, Gall and Inglis, Edinburgh, 1969
F. M. Jackson, Polytopes in Euclidean n-Space. Inst. Math. Appl. Bull. (UK) 29, 172-174, Nov./Dec. 1993.
Eric W. Weisstein, Spherical Excess at MathWorld.
Eric W. Weisstein, Solid Angle at MathWorld.

Categories: Angle | Geometry

Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Results from FactBites:

Solid angle - Wikipedia, the free encyclopedia (496 words)
	Thus the solid angle of a sphere measured at its center is 4π steradians, and the solid angle subtended at the center of a cube by one of its sides is one-sixth of this, that is, 2π/3 steradians.
	A solid angle is the three dimensional analogue of the ordinary angle, and the steradian is similarly analogous to the radian.
	A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs on a sphere, and is measured by the angle between the planes containing the arcs and the centre of the sphere.

More results at FactBites »

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