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Solid angle

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	Topics Solid angle Topic Home 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 areaArea (geometry) Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface.... , S, of a projection of that object onto a sphereSphere A sphere is a perfectly symmetrical geometrical object.... centered at that point, divided by 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 angleAngle Summary An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.... is related to the circumferenceCircumference The circumference is the distance around a closed curve.... of a circleCircle In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed poi... . If the proportionality constant is chosen to be 1, the units of solid angle will be the SISi Si, si, or SI may stand for:... steradianSteradian The steradian is the SI unit of solid angle.... (abbreviated "sr"). Thus the solid angle of a sphere measured from a point in its interior is 4pPi The mathematical constant p is an irrational real number, approximately equal to 3.14159, which is the ratio of a circle's c... sr, and the solid angle subtended at the center of a cube by one of its sides is one-sixth of that, or 2p/3 sr. Solid angles can also be measured (for k = (180/p)²) in square degreeSquare degree A square degree is a non-SI unit that can be used to measure solid angles.... 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: To obtain the solid angle in steradians, multiply the fractional area by 4p. 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 vectorVector (spatial) In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a... position of an infinitesimal area of surface with respect to point P and where represents the unit vector normal to . Practical applications Defining luminous intensityLuminous intensity In photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular di... and luminanceLuminance Luminance is a photometric measure of the density of luminous intensity in a given direction.... Calculating spherical excess E of a spherical triangle The calculation of potentials by using the boundary element methodBoundary element method The boundary element method is a numerical computational method of solving linear partial differential equations which have ... (BEM) Evaluating the size of ligandLigand In chemistry, a ligand is an atom, ion, or molecule that generally donates one or more of its electrons through a coordinat... s in metal complexes, see ligand cone angleLigand cone angle Ligand cone angle is a measure of the size of a ligand.... . Calculating the electric fieldElectric field In physics, the properties of space that surrounds an electric charge can be described using an electric field or E-field... and magnetic fieldMagnetic field Summary In physics, a magnetic field is that part of the electromagnetic field that exists when there is a changing electric field.... strength around charge distributions. Solid angles for common objects Cone, spherical cap, hemisphere The solid angle of a coneCone (geometry) In common usage and elementary geometry, a cone is a solid object obtained by rotating a right triangle around one of its tw... with apexApex (geometry) In geometry, an apex is a descriptive label for a visual singular highest point or vertex in a triangle, pyramid or cone, us... angle , is the area of a spherical capSpherical cap In geometry, a spherical cap is a portion of a sphere cut off by a plane.... on a unit sphere (The above result is found by computing the following double integralDouble integral In mathematical analysis, there is an important distinction between a double integral and an iterated integral.... using the unit surface element in spherical polarsFacts About Volume and surface elements in different co-ordinate systems This page outlines the value of different volume and surface elements in several different co-ordinate systems.... ): Over 2200 years ago ArchimedesArchimedes Archimedes was an ancient Greek mathematician, physicist, engineer, astronomer, and philosopher born in the seaport colony... proved, without the use of calculusCalculus Calculus is a central branch of mathematics, developed from algebra and geometry.... , 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: Hence for a unit sphere the solid angle of the spherical cap is given as: When ? = p/2, the spherical cap becomes a hemisphereSphere A sphere is a perfectly symmetrical geometrical object.... having a solid angle 2p. Pyramid The solid angle of a four-sided right rectangular pyramidPyramid Pyramids are among the largest man-made constructions as well as one of the great Wonders of the ancient world.... with apex angles and 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 apexApex (geometry) In geometry, an apex is a descriptive label for a visual singular highest point or vertex in a triangle, pyramid or cone, us... of the pyramid (the center of the sfere) are known, then the above equation can be manipulated to give Latitude-longitude rectangle The solid angle of a latitude-longitude rectangle on a globeGlobe This article is on a planet-representation device.... is , where and are north and south lines of latitudeLatitude Latitude, usually denoted symbolically by the Greek letter f , gives the location of a place on Earth north or south of the ... (measured from the equatorEquator The equator is an imaginary circle drawn around a planet at a distance halfway between the poles.... in radians with angle increasing northward), and and are east and west lines of longitudeLongitude Longitude, sometimes denoted by the Greek letter ? , describes the location of a place on Earth east or west of a north-sout... (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 circleGreat circle A great circle is a circle on the surface of a sphere that has the same circumference as the sphere, and divides 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 SunSun \|+ The Sun \|+\|-\| colspan="2" align="center" \| \|-... and MoonMoon The Moon is Earth's only natural satellite.... 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 functionGamma function In mathematics, the Gamma function extends the factorial function to complex and non-integer numbers .... . Since 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 sradSteradian The steradian is the SI unit of solid angle.... 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 measureMeasure (mathematics) In mathematics, a measure is a function that assigns a number, e.g., a "size", "volume", or "probability", to subsets of a g... of 2.