solid angle

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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, a solid angle (symbol: ) is a measure of the amount of the
field of view The field of view (FoV) is the extent of the observable world that is seen at any given moment. In the case of optical instruments or sensors it is a solid angle through which a detector is sensitive to electromagnetic radiation. Humans ...
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 The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
(SI), a solid angle is expressed in a
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1 ...
unit called a ''
steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radia ...
'' (symbol: sr). One steradian corresponds to one unit of area on the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A uni ...
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 The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of a ...
of the unit sphere, $4\pi$. 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 The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width o ...
is much smaller than the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, it is also much closer to
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large list of largest lakes and seas in the Solar System, volumes of water can be found throughout the Solar System, only water distributi ...
. 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 A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby obscuring the view of the Sun from a small part of the Earth, totally or partially. Such an alignment occurs during an eclipse season, approximately every six mon ...
.

Definition and properties

An object's solid angle in
steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radia ...
s is equal to the
area Area is the quantity that expresses the extent of a Region (mathematics), region on the plane (geometry), plane or on a curved surface (mathematics), surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar ...
of the segment of a
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A uni ...
, 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 like a planar angle in radians is the ratio of the length of an arc to its radius, a solid angle in steradians is the ratio of the area covered on a sphere by an object to the area given by the square of the radius of said sphere. The formula is $\Omega=\frac,$ where A is the spherical surface area and r is the radius of the considered sphere. Solid angles are often used in
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxie ...
,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rela ...
, and in particular
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the hea ...
. 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, and the solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or  sr. Solid angles can also be measured in
square degree __NOTOC__ A square degree (deg2) is a non- SI unit measure of solid angle. Other denotations include ''sq. deg.'' and (°)2. Just as degrees are used to measure parts of a circle, square degrees are used to measure parts of a sphere. Analogous to ...
s (1 sr = 2 square degrees), in square minutes and square seconds, or in fractions of the sphere (1 sr = fractional area), also known as spat (1 sp = 4 sr). In
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
there is a formula for the differential, $d\Omega = \sin\theta\,d\theta\,d\varphi,$ where is the
colatitude In a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude. Here Southern latitudes are defined to be negative, and as a result the colatitude is a non ...
(angle from the North Pole) and is the longitude. The solid angle for an arbitrary
oriented surface In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
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 In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
: $\Omega = \iint_S \frac\,dS \ = \iint_S \sin\theta\,d\theta\,d\varphi,$ where $\hat = \vec / r$ is the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
corresponding to $\vec$, the
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
of an infinitesimal area of surface with respect to point , and where $\hat$ represents the unit
normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
to . Even if the projection on the unit sphere to the surface is not
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product $\hat \cdot \hat$. Thus one can approximate the solid angle subtended by a small facet having flat surface area , orientation $\hat$, and distance from the viewer as: $d\Omega = 4 \pi \left(\frac\right) \, (\hat \cdot \hat),$ where the surface area of a sphere is .

Practical applications

*Defining
luminous intensity In photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human eye ...
and
luminance Luminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through, is emitted from, or is reflected from a particular area, and falls withi ...
*Calculating
spherical excess Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
of a
spherical triangle Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
*The calculation of potentials by using the boundary element method (BEM) *Evaluating the size of
ligand In coordination chemistry, a ligand is an ion or molecule (functional group) that binds to a central metal atom to form a coordination complex. The bonding with the metal generally involves formal donation of one or more of the ligand's elect ...
s in metal complexes, see ligand cone angle *Calculating the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field f ...
and
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
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 Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrational energy being gained by a ...
*The solid angle of the acceptance cone of the
optical fiber An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means ...

Solid angles for common objects

Cone, spherical cap, hemisphere

The solid angle of a
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines co ...
with its apex at the apex of the solid angle, and with
apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Ape ...
angle 2, is the area of a spherical cap on a
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A uni ...
$\Omega = 2\pi \left (1 - \cos\theta \right)\ = 4\pi \sin^2 \frac.$ For small such that this reduces to , the area of a circle. The above is found by computing the following
double integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-numbe ...
using the unit surface element in spherical coordinates: This formula can also be derived without the use of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...
. Over 2200 years ago
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists i ...
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. In the diagram this radius is given as $2r \sin \frac.$ Hence for a unit sphere the solid angle of the spherical cap is given as $\Omega = 4\pi \sin^2 \frac = 2\pi \left (1 - \cos\theta \right).$ When = , the spherical cap becomes a hemisphere having a solid angle 2. The solid angle of the complement of the cone is $4\pi - \Omega = 2\pi \left(1 + \cos\theta \right) = 4\pi\cos^2 \frac.$ This is also the solid angle of the part of the
celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface 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 For example, if , then the formula reduces to the spherical cap formula above: the first term becomes , and the second .

Tetrahedron

Let OABC be the vertices of a
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
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 to be the angle BOC and define , correspondingly. Let $\phi_$ be the
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
between the planes that contain the tetrahedral faces OAC and OBC and define $\phi_$, $\phi_$ correspondingly. The solid angle subtended by the triangular surface ABC is given by $\Omega = \left(\phi_ + \phi_ + \phi_\right)\ - \pi.$ This follows from the theory of
spherical excess Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
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: $\sum_^4 \Omega_i = 2 \sum_^6 \phi_i\ - 4 \pi,$ 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. 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 $\tan \left( \frac \Omega \right) = \sqrt,$ where $\theta_s = \frac .$ Another interesting formula involves expressing the vertices as vectors in 3 dimensional space. Let $\vec a\ ,\, \vec b\ ,\, \vec c$ 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: $\tan \left( \frac \Omega \right) = \frac,$ where $\left, \vec a\ \vec b\ \vec c\=\vec a \cdot (\vec b \times \vec c)$ denotes the
scalar triple product In geometry and algebra, the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector ...
of the three vectors and $\vec a \cdot \vec b$ 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 winding. Computing 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 .

Pyramid

The solid angle of a four-sided right rectangular
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
with
apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Ape ...
angles and (
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
s measured to the opposite side faces of the pyramid) is $\Omega = 4 \arcsin \left( \sin \left(\right) \sin \left(\right) \right).$ 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 $\Omega = 4 \arctan \frac .$ The solid angle of a right -gonal pyramid, where the pyramid base is a regular -sided polygon of circumradius , with a pyramid height is $\Omega = 2\pi - 2n \arctan\left(\frac \right).$ 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: where parentheses (* *) is a scalar product and square brackets * *is a
scalar triple product In geometry and algebra, the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector ...
, and is an
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. Indices are cycled: and . The complex products add the phase associated with each vertex angle of the polygon. However, a multiple of $2\pi$ is lost in the branch cut of $\arg$ 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 rectangle

The solid angle of a latitude-longitude rectangle on a globe is $\left ( \sin \phi_\mathrm - \sin \phi_\mathrm \right ) \left ( \theta_\mathrm - \theta_\mathrm \,\! \right)\;\mathrm,$ where and are north and south lines of
latitude In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
(measured from the
equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can a ...
in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s with angle increasing northward), and and are east and west lines of
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lett ...
(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 In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry ...
arcs. With a latitude-longitude rectangle, only lines of longitude are great circle arcs; lines of latitude are not.

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, $d$: $\Omega = 2 \pi \left (1 - \frac \right ) : d \geq R.$ By inputting the appropriate average values for the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
and the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width o ...
(in relation to Earth), the average solid angle of the Sun is steradians and the average solid angle of the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width o ...
is steradians. In terms of the total celestial sphere, the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
and the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width o ...
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
eclipses An eclipse is an astronomical event that occurs when an astronomical object or spacecraft is temporarily obscured, by passing into the shadow of another body or by having another body pass between it and the viewer. This alignment of three ce ...
depending on the distance between the Earth and the Moon during the eclipse.

Solid angles in arbitrary dimensions

The solid angle subtended by the complete ()-dimensional spherical surface of the unit sphere in -dimensional 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 $\Omega_ = \frac,$ where is the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. When is an integer, the gamma function can be computed explicitly. It follows that $\Omega_ = \begin \frac 2\pi^\frac\ & d\text \\ \frac 2^d \pi^\ & d\text. \end$ 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: Given unit vectors $\vec_i$ defining the angle, let denote the matrix formed by combining them so the th column is $\vec_i$, and $\alpha_ = \vec_i\cdot\vec_j = \alpha_, \alpha_=1$. The variables $\alpha_,1 \le i < j \le d$ form a multivariable $\vec \alpha = \left(\alpha_,\dotsc , \alpha_, \alpha_, \dotsc, \alpha_\right) \in \R^$. For a "congruent" integer multiexponent $\vec a=\left(a_, \dotsc, a_, a_, \dotsc , a_\right) \in \N_0^,$ define $\vec \alpha^=\prod \alpha_^$. Note that here $\N_0$ = non-negative integers, or natural numbers beginning with 0. The notation $\alpha_$ for $j > i$ means the variable $\alpha_$, similarly for the exponents $a_$. Hence, the term $\sum_ a_$ means the sum over all terms in $\vec a$ 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.