Solid angle

The solid angle, Ω, is the angle in three-dimensional space that an object subtends at a point. In Geometry, an arc subtended by an Angle is a Curve whose endpoints are on the angle's two rays 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. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe (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 Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the

If the proportionality constant is chosen to be 1, the units of solid angle will be the SI steradian (abbreviated "sr"). The steradian (symbol sr) is the SI unit of Solid angle. It is used to describe two-dimensional angular spans in three- Dimensional space 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. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 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. A square degree (denoted deg²) is a non- SI unit measure of Solid angle. e. , fractional area).

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 $\, dS$ .

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. In photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a Light source in a particular direction per unit Solid Luminance is a photometric measure of the density of Luminous intensity in a given direction Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations 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 either an Atom, Ion, or Molecule (see also Functional group) that bonds to a central metal generally Ligand cone angle (also known as the Tolman cone angle) is a measure of the size of a Ligand.
Calculating the electric field and magnetic field strength around charge distributions. 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 (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges

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. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. 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. In Aerospace engineering, the Dihedral is the Angle between the two wings see Dihedral. The solid angle at O subtended by the triangular surface ABC is given by

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

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:

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

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. Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations

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. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n g. $M_{i1}=\vec a_i$ and so on--this is also equivalent to the scalar triple product of the three 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. This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R

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

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

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. Simon Antoine Jean L'Huilier (or L'Huillier) ( Geneva, 24 April 1750 - Geneva, 28 March 1840) was a Swiss 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

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

(The above result is found by computing the following double integral using the unit surface element in spherical polars):

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

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. A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex In Geometry, a spherical cap is a portion of a Sphere cut off by a plane. In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed The multiple integral is a type of definite Integral extended to functions of more than one real Variable, for example This page outlines the value of different volume and Surface elements in several different Coordinate systems See also Line integral Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In the diagram opposite this radius is given as:

$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. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

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 $4 \arcsin \left (\sin {a \over 2} \sin {b \over 2} \right). \,\!$

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

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

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). A pyramid is a Building where the upper surfaces are triangular and converge on one point In Aerospace engineering, the Dihedral is the Angle between the two wings see Dihedral. A globe is a three- Dimensional scale model of Earth ( terrestrial globe) or other spheroid celestial body such as a planet star or moon Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the The equator (sometimes referred to colloquially as "the Line") is the intersection of the Earth 's surface with the plane perpendicular to the The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement [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.

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. A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. With a latitude-longitude rectangle, only lines of longitude are great circle arcs; lines of latitude are not.

Sun and Moon

The Sun and Moon are both seen from Earth at a fractional area of 0. The Sun (Sol is the Star at the center of the Solar System. 001% of the celestial hemisphere or about 6×10^-5 steradian. [2]

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 $d$ . In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula

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

where $Γ$ is the Gamma function. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function Since $d$ is an integer, the Gamma function can be computed explicitly. It follows that

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

This gives the expected results of 2π rad for the 2D circumference and 4π srad for the 3D sphere. The steradian (symbol sr) is the SI unit of Solid angle. It is used to describe two-dimensional angular spans in three- Dimensional space 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. 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

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 (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Eric W. Weisstein, Solid Angle at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

Dictionary

-noun

(geometry) The three -dimensional analog of an angle.

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