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Solid angle
 
 
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The solid angle, O, 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
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 centered at that point, and inversely proportional to the square of the sphere's radius, R.






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The solid angle, O, 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
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 centered at that point, and inversely proportional to the square of the sphere's radius, R. (Symbolically, O = 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
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
 is related to the circumference
Circumference

The circumference is the distance around a closed curve. Circumference is a kind of perimeter....
 of a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
.
Steradian
If the proportionality constant is chosen to be 1, the units of solid angle will be the SI
Si

Si, si, or SI may refer to :...
 steradian
Steradian

The steradian is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a Plane ....
 (abbreviated "sr"). Thus the solid angle of a sphere measured from a point in its interior is 4p
Pi

Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
 sr, and the solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2p/3 sr. Solid angles can also be measured (for k = (180/p)2) in square degree
Square degree

A square degree is a non-SI Units of measurement measure of solid angle. Just as degree s are used to measure parts of a circle, square degrees are used to measure parts of a sphere....
s or (for k = 1/4p) in fractions of the sphere (i.e., fractional area).(sr) 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:

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


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

where is the vector position of an infinitesimal area of surface with respect to point P and where represents the unit vector normal to .

Practical applications

  • Defining luminous intensity
    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

    Luminance is a Photometry measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle....
  • Calculating spherical excess E of a spherical triangle
  • The calculation of potentials by using the boundary element method
    Boundary element method

    The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations ....
     (BEM)
  • Evaluating the size of ligand
    Ligand

    In chemistry, a ligand is either an atom, ion, or molecule that bonds to a central metal, generally involving formal donation of one or more of its electrons....
    s in metal complexes, see ligand cone angle
    Ligand cone angle

    Ligand cone angle is a measure of the size of a ligand. It is defined as the solid angle formed with the metal at the vertex and the hydrogen atoms at the perimeter of the cone ....
    .
  • Calculating the electric field
    Electric field

    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 ....
     and magnetic field
    Magnetic field

    A magnetism field is a vector field which can exert a magnetic force on moving electric charges and on magnetic dipoles . When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field....
     strength around charge distributions.
  • Deriving Gauss's Law
    Gauss's law

    In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field....
    .


Solid angles for common objects


Tetrahedron

Let OABC be the vertices of a tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
 with an origin at O subtended by the triangular face ABC where are the vector positions of the vertices A, B and C. Define the vertex angle to be the angle BOC and define correspondingly. Let be the dihedral angle
Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
 between the planes that contain the tetrahedral faces OAC and OBC and define correspondingly. The solid angle at O subtended by the triangular surface ABC is given by

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:

where 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 are the vector positions of the vertices A, B and C has been given by Oosterom and Strackee:

where



denotes the determinant
Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
 of the matrix that results when writing the vectors together in a row, e.g. and so on--this is also equivalent to the scalar triple product
Triple product

In vector calculus, there are two ways of multiplying three vector together, to make a triple product of vectors.Scalar triple product ...
 of the three vectors;

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


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 is given by L' Huilier
Simon Antoine Jean L'Huilier

Simon Antoine Jean L'Huilier was a Swiss mathematician of France Hugenot descent. He is known for his work in mathematical analysis and topology, and in particular the generalization of Euler characteristic for planar graphs....
's theorem as



where


Cone, spherical cap, hemisphere

The solid angle of a cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
 with apex
Apex (geometry)

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....
 angle , is the area of a spherical cap
Spherical cap

In geometry, a spherical cap is a portion of a sphere cut off by a Plane . If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere....
 on a unit sphere

(The above result is found by computing the following double integral using the unit surface element in spherical polars
Volume and surface elements in different co-ordinate systems

This page outlines the value of different volume element and surface elements in several different coordinate systems.Surface elements depend on the orientation of the surface over which the integration is performed....
):



Over 2200 years ago Archimedes
Archimedes

Archimedes of Syracuse was a Greek mathematics, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity....
 proved, without the use of calculus
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
, that the surface area of a spherical cap was always equal to the area of a circle whose radius was equal to 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 opposite this radius is given as:

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

When , the spherical cap becomes a hemisphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 having a solid angle 2p.

Pyramid

The solid angle of a four-sided right rectangular pyramid
Pyramid

A pyramid is a building where the outer surfaces are triangular and converge at a point. The base of pyramids are usually quadrilateral or trilateral , meaning that a pyramid usually has four or five faces....
 with apex angles and (dihedral angle
Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
s measured to the opposite side faces of the pyramid) is


If both the side lengths (a and ß) of the base of the pyramid and the distance (d) from the center of the base rectangle to the apex
Apex (geometry)

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....
 of the pyramid (the center of the sphere) are known, then the above equation can be manipulated to give



Latitude-longitude rectangle

The solid angle of a latitude-longitude rectangle on a globe
Globe

A globe is a three-dimensional scale Model of Earth or other spheroid celestial body such as a planet, star, or moon. It may also refer to a spherical representation of the celestial sphere, showing the apparent positions of the stars in the sky ...
 is , where and are north and south lines of latitude
Latitude

Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
 (measured from the equator
Equator

The equator is the intersection of the Earth's surface with the Plane perpendicular to the Earth's rotation and containing the Earth's center of mass....
 in radians with angle increasing northward), and and are east and west lines of longitude
Longitude

Longitude , symbolized by the Greek character lambda , is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement....
 (where the angle in radians increases eastward). Mathematically, this represents an arc of angle swept around a sphere by radians. When longitude spans 2p radians and latitude spans p 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
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
 arcs. With a latitude-longitude rectangle, only lines of longitude are great circle arcs; lines of latitude are not.

Sun and Moon

The Sun
Sun

The Sun , a G V star, is the star at the center of the Solar System. The Earth and other matter orbit the Sun, which by itself accounts for about 98.6% of the Solar System's mass....
 and Moon
Moon

The Moon is Earth's only natural satellite and the List of natural satellites by diameter satellite in the Solar System. The average centre-to-centre distance from the Earth to the Moon is km, about thirty times the diameter of the Earth....
 are both seen from Earth at a fractional area of 0.001% of the celestial hemisphere or about 6 steradian.

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 . One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula

where is the Gamma function
Gamma function

In mathematics, the Gamma function is an extension of the factorial function to real number and complex number numbers. For a complex number z with positive real part the Gamma function is defined by...
. When is an integer, the Gamma function can be computed explicitly. It follows that

This gives the expected results of 2p rad for the 2D circumference and 4p sr
Steradian

The steradian is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a Plane ....
 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 , which indeed has a measure
Measure (mathematics)

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset....
 of 2.