[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/solid-angle-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/solid-angle-wikipedia\/","headline":"Solid angle – Wikipedia","name":"Solid angle – Wikipedia","description":"Measure of how large an object appears to an observer at a given point in three-dimensional space In geometry, a","datePublished":"2019-06-06","dateModified":"2019-06-06","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/057444bf35a0c22b19bcae1ef06e06ecdf8abe56","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/057444bf35a0c22b19bcae1ef06e06ecdf8abe56","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/solid-angle-wikipedia\/","about":["Wiki"],"wordCount":19531,"articleBody":"Measure of how large an object appears to an observer at a given point in three-dimensional spaceIn geometry, a solid angle (symbol: \u03a9) is a measure of the amount of the field of view 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 (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr). One steradian corresponds to one unit of area on the unit sphere 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 of the unit sphere, 4\u03c0{displaystyle 4pi }. 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 is much smaller than the Sun, it is also much closer to Earth. 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.Table of ContentsDefinition and properties[edit]Practical applications[edit]Solid angles for common objects[edit]Cone, spherical cap, hemisphere[edit]Tetrahedron[edit]Pyramid[edit]Latitude-longitude rectangle[edit]Celestial objects[edit]Solid angles in arbitrary dimensions[edit]References[edit]Further reading[edit]External links[edit]Definition and properties[edit]An object’s solid angle in steradians is equal to the area of the segment of a unit sphere, 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\u03a9=Ar2,{displaystyle Omega ={frac {A}{r^{2}}},}where A{displaystyle A} is the spherical surface area and r{displaystyle r} is the radius of the considered sphere.Solid angles are often used in astronomy, physics, and in particular astrophysics. 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. Any area on a sphere which is equal in area to the square of its radius, when observed from its center, subtends precisely one steradian.The solid angle of a sphere measured from any point in its interior is 4\u03c0\u00a0sr, and the solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2\u03c0\/3\u00a0 sr. Solid angles can also be measured in square degrees (1\u00a0sr = (180\/\u03c0)2 square degrees), in square minutes and square seconds, or in fractions of the sphere (1\u00a0sr = 1\/4\u03c0 fractional area), also known as spat (1\u00a0sp = 4\u03c0\u00a0sr).In spherical coordinates there is a formula for the differential,d\u03a9=sin\u2061\u03b8d\u03b8d\u03c6,{displaystyle dOmega =sin theta ,dtheta ,dvarphi ,}where \u03b8 is the colatitude (angle from the North Pole) and \u03c6 is the longitude.The solid angle for an arbitrary oriented surface S subtended at a point P is equal to the solid angle of the projection of the surface S to the unit sphere with center P, which can be calculated as the surface integral:\u03a9=\u222cSr^\u22c5n^r2dS\u00a0=\u222cSsin\u2061\u03b8d\u03b8d\u03c6,{displaystyle Omega =iint _{S}{frac {{hat {r}}cdot {hat {n}}}{r^{2}}},dS =iint _{S}sin theta ,dtheta ,dvarphi ,}where r^=r\u2192\/r{displaystyle {hat {r}}={vec {r}}\/r} is the unit vector corresponding to r\u2192{displaystyle {vec {r}}}, the position vector of an infinitesimal area of surface dS with respect to point P, and where n^{displaystyle {hat {n}}} represents the unit normal vector to dS. Even if the projection on the unit sphere to the surface S is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product r^\u22c5n^{displaystyle {hat {r}}cdot {hat {n}}}.Thus one can approximate the solid angle subtended by a small facet having flat surface area dS, orientation n^{displaystyle {hat {n}}}, and distance r from the viewer as:d\u03a9=4\u03c0(dSA)(r^\u22c5n^),{displaystyle dOmega =4pi left({frac {dS}{A}}right),({hat {r}}cdot {hat {n}}),}where the surface area of a sphere is A = 4\u03c0r2.Practical applications[edit]Solid angles for common objects[edit]Cone, spherical cap, hemisphere[edit] Section of cone (1) and spherical cap (2) inside a sphere. In this figure \u03b8 = A\/2 and r = 1.The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2\u03b8, is the area of a spherical cap on a unit sphere\u03a9=2\u03c0(1\u2212cos\u2061\u03b8)\u00a0=4\u03c0sin2\u2061\u03b82.{displaystyle Omega =2pi left(1-cos theta right) =4pi sin ^{2}{frac {theta }{2}}.}For small \u03b8 such that cos \u03b8 \u2248 1 \u2212 \u03b82\/2 this reduces to \u03c0\u03b82, the area of a circle.The above is found by computing the following double integral using the unit surface element in spherical coordinates:\u222b02\u03c0\u222b0\u03b8sin\u2061\u03b8\u2032d\u03b8\u2032d\u03d5=\u222b02\u03c0d\u03d5\u222b0\u03b8sin\u2061\u03b8\u2032d\u03b8\u2032=2\u03c0\u222b0\u03b8sin\u2061\u03b8\u2032d\u03b8\u2032=2\u03c0[\u2212cos\u2061\u03b8\u2032]0\u03b8=2\u03c0(1\u2212cos\u2061\u03b8).{displaystyle {begin{aligned}int _{0}^{2pi }int _{0}^{theta }sin theta ‘,dtheta ‘,dphi &=int _{0}^{2pi }dphi int _{0}^{theta }sin theta ‘,dtheta ‘\\&=2pi int _{0}^{theta }sin theta ‘,dtheta ‘\\&=2pi left[-cos theta ‘right]_{0}^{theta }\\&=2pi left(1-cos theta right).end{aligned}}}This formula can also be derived without the use of calculus. Over 2200 years ago Archimedes 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.[1] In the diagram this radius is given as2rsin\u2061\u03b82.{displaystyle 2rsin {frac {theta }{2}}.}Hence for a unit sphere the solid angle of the spherical cap is given as\u03a9=4\u03c0sin2\u2061\u03b82=2\u03c0(1\u2212cos\u2061\u03b8).{displaystyle Omega =4pi sin ^{2}{frac {theta }{2}}=2pi left(1-cos theta right).}When \u03b8 = \u03c0\/2, the spherical cap becomes a hemisphere having a solid angle 2\u03c0.The solid angle of the complement of the cone is4\u03c0\u2212\u03a9=2\u03c0(1+cos\u2061\u03b8)=4\u03c0cos2\u2061\u03b82.{displaystyle 4pi -Omega =2pi left(1+cos theta right)=4pi cos ^{2}{frac {theta }{2}}.}This is also the solid angle of the part of the celestial sphere that an astronomical observer positioned at latitude \u03b8 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 \u03b3 from the cone’s axis and passing through the cone’s apex can be calculated by the formula[2]\u03a9=2[arccos\u2061(sin\u2061\u03b3sin\u2061\u03b8)\u2212cos\u2061\u03b8arccos\u2061(tan\u2061\u03b3tan\u2061\u03b8)].{displaystyle Omega =2left[arccos left({frac {sin gamma }{sin theta }}right)-cos theta arccos left({frac {tan gamma }{tan theta }}right)right].}For example, if \u03b3 = \u2212\u03b8, then the formula reduces to the spherical cap formula above: the first term becomes \u03c0, and the second \u03c0 cos \u03b8.Tetrahedron[edit]Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where a\u2192\u00a0,b\u2192\u00a0,c\u2192{displaystyle {vec {a}} ,,{vec {b}} ,,{vec {c}}} are the vector positions of the vertices A, B and C. Define the vertex angle \u03b8a to be the angle BOC and define \u03b8b, \u03b8c correspondingly. Let \u03d5ab{displaystyle phi _{ab}} be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define \u03d5ac{displaystyle phi _{ac}}, \u03d5bc{displaystyle phi _{bc}} correspondingly. The solid angle \u03a9 subtended by the triangular surface ABC is given by\u03a9=(\u03d5ab+\u03d5bc+\u03d5ac)\u00a0\u2212\u03c0.{displaystyle Omega =left(phi _{ab}+phi _{bc}+phi _{ac}right) -pi .}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 \u03c0“, for the sum of the four internal solid angles of a tetrahedron as follows:\u2211i=14\u03a9i=2\u2211i=16\u03d5i\u00a0\u22124\u03c0,{displaystyle sum _{i=1}^{4}Omega _{i}=2sum _{i=1}^{6}phi _{i} -4pi ,}where \u03d5i{displaystyle phi _{i}} ranges over all six of the dihedral angles between any two planes that contain the tetrahedral faces OAB, OAC, OBC and ABC.[3]A useful formula for calculating the solid angle of the tetrahedron at the origin O that is purely a function of the vertex angles \u03b8a, \u03b8b, \u03b8c is given by L’Huilier’s theorem[4][5] astan\u2061(14\u03a9)=tan\u2061(\u03b8s2)tan\u2061(\u03b8s\u2212\u03b8a2)tan\u2061(\u03b8s\u2212\u03b8b2)tan\u2061(\u03b8s\u2212\u03b8c2),{displaystyle 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\u03b8s=\u03b8a+\u03b8b+\u03b8c2.{displaystyle theta _{s}={frac {theta _{a}+theta _{b}+theta _{c}}{2}}.}Another interesting formula involves expressing the vertices as vectors in 3 dimensional space. Let a\u2192\u00a0,b\u2192\u00a0,c\u2192{displaystyle {vec {a}} ,,{vec {b}} ,,{vec {c}}} be the vector positions of the vertices A, B and C, and let a, b, and c be the magnitude of each vector (the origin-point distance). The solid angle \u03a9 subtended by the triangular surface ABC is:[6][7]tan\u2061(12\u03a9)=|a\u2192\u00a0b\u2192\u00a0c\u2192|abc+(a\u2192\u22c5b\u2192)c+(a\u2192\u22c5c\u2192)b+(b\u2192\u22c5c\u2192)a,{displaystyle tan left({frac {1}{2}}Omega right)={frac {left|{vec {a}} {vec {b}} {vec {c}}right|}{abc+left({vec {a}}cdot {vec {b}}right)c+left({vec {a}}cdot {vec {c}}right)b+left({vec {b}}cdot {vec {c}}right)a}},}where|a\u2192\u00a0b\u2192\u00a0c\u2192|=a\u2192\u22c5(b\u2192\u00d7c\u2192){displaystyle left|{vec {a}} {vec {b}} {vec {c}}right|={vec {a}}cdot ({vec {b}}times {vec {c}})}denotes the scalar triple product of the three vectors and a\u2192\u22c5b\u2192{displaystyle {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 a, b, c have the wrong winding. Computing the absolute value 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 \u03c0.Pyramid[edit]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\u03a9=4arcsin\u2061(sin\u2061(a2)sin\u2061(b2)).{displaystyle Omega =4arcsin left(sin left({a over 2}right)sin left({b over 2}right)right).}If both the side lengths (\u03b1 and \u03b2) 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 sphere) are known, then the above equation can be manipulated to give\u03a9=4arctan\u2061\u03b1\u03b22d4d2+\u03b12+\u03b22.{displaystyle Omega =4arctan {frac {alpha beta }{2d{sqrt {4d^{2}+alpha ^{2}+beta ^{2}}}}}.}The solid angle of a right n-gonal pyramid, where the pyramid base is a regular n-sided polygon of circumradius r, with apyramid height h is\u03a9=2\u03c0\u22122narctan\u2061(tan\u2061(\u03c0n)1+r2h2).{displaystyle Omega =2pi -2narctan left({frac {tan left({pi over n}right)}{sqrt {1+{r^{2} over h^{2}}}}}right).}The solid angle of an arbitrary pyramid with an n-sided base defined by the sequence of unit vectors representing edges {s1, s2}, … sn can be efficiently computed by:[2]\u03a9=2\u03c0\u2212arg\u2061\u220fj=1n((sj\u22121sj)(sjsj+1)\u2212(sj\u22121sj+1)+i[sj\u22121sjsj+1]).{displaystyle Omega =2pi -arg prod _{j=1}^{n}left(left(s_{j-1}s_{j}right)left(s_{j}s_{j+1}right)-left(s_{j-1}s_{j+1}right)+ileft[s_{j-1}s_{j}s_{j+1}right]right).}where parentheses (* *) is a scalar product and square brackets [* * *] is a scalar triple product, and i is an imaginary unit. Indices are cycled: s0 = sn and s1 = sn + 1. The complex products add the phase associated with each vertex angle of the polygon. However, a multiple of2\u03c0{displaystyle 2pi } is lost in the branch cut of arg{displaystyle 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[edit]The solid angle of a latitude-longitude rectangle on a globe is(sin\u2061\u03d5N\u2212sin\u2061\u03d5S)(\u03b8E\u2212\u03b8W)sr,{displaystyle left(sin phi _{mathrm {N} }-sin phi _{mathrm {S} }right)left(theta _{mathrm {E} }-theta _{mathrm {W} },!right);mathrm {sr} ,}where \u03c6N and \u03c6S are north and south lines of latitude (measured from the equator in radians with angle increasing northward), and \u03b8E and \u03b8W are east and west lines of longitude (where the angle in radians increases eastward).[8] Mathematically, this represents an arc of angle \u03d5N \u2212 \u03d5S swept around a sphere by \u03b8E \u2212 \u03b8W radians. When longitude spans 2\u03c0 radians and latitude spans \u03c0 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. With a latitude-longitude rectangle, only lines of longitude are great circle arcs; lines of latitude are not.Celestial objects[edit]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{textstyle R}, and the distance from the observer to the object, d{displaystyle d}:\u03a9=2\u03c0(1\u2212d2\u2212R2d):d\u2265R.{displaystyle Omega =2pi left(1-{frac {sqrt {d^{2}-R^{2}}}{d}}right):dgeq R.}By inputting the appropriate average values for the Sun and the Moon (in relation to Earth), the average solid angle of the Sun is 6.794\u00d710\u22125 steradians and the average solid angle of the Moon is 6.418\u00d710\u22125 steradians. In terms of the total celestial sphere, the Sun and the Moon subtend average fractional areas of 0.0005406% (5.406\u00a0ppm) and 0.0005107% (5.107\u00a0ppm), respectively. As these solid angles are about the same size, the Moon can cause both total and annular solar eclipses depending on the distance between the Earth and the Moon during the eclipse.Solid angles in arbitrary dimensions[edit]The solid angle subtended by the complete (d \u2212 1)-dimensional spherical surface of the unit sphere in d-dimensional Euclidean space 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\u03a9d=2\u03c0d2\u0393(d2),{displaystyle Omega _{d}={frac {2pi ^{frac {d}{2}}}{Gamma left({frac {d}{2}}right)}},}where \u0393 is the gamma function. When d is an integer, the gamma function can be computed explicitly.[9] It follows that\u03a9d={1(d2\u22121)!2\u03c0d2\u00a0d\u00a0even(12(d\u22121))!(d\u22121)!2d\u03c012(d\u22121)\u00a0d\u00a0odd.{displaystyle Omega _{d}={begin{cases}{frac {1}{left({frac {d}{2}}-1right)!}}2pi ^{frac {d}{2}} &d{text{ even}}\\{frac {left({frac {1}{2}}left(d-1right)right)!}{(d-1)!}}2^{d}pi ^{{frac {1}{2}}(d-1)} &d{text{ odd}}.end{cases}}}This gives the expected results of 4\u03c0 steradians for the 3D sphere bounded by a surface of area 4\u03c0r2 and 2\u03c0 radians for the 2D circle bounded by a circumference of length 2\u03c0r. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D “sphere” is the interval [\u2212r, r] and this is bounded by two limiting points.The counterpart to the vector formula in arbitrary dimension was derived by Aomoto[10][11]and independently by Ribando.[12] It expresses them as an infinite multivariate Taylor series:\u03a9=\u03a9d|det(V)|(4\u03c0)d\/2\u2211a\u2192\u2208N0(d2)[(\u22122)\u2211ii\u0393(1+\u2211m\u2260iaim2)]\u03b1\u2192a\u2192.{displaystyle Omega =Omega _{d}{frac {left|det(V)right|}{(4pi )^{d\/2}}}sum _{{vec {a}}in mathbb {N} _{0}^{binom {d}{2}}}left[{frac {(-2)^{sum _{ii{displaystyle {vec {v}}_{i}}, and \u03b1ij=v\u2192i\u22c5v\u2192j=\u03b1ji,\u03b1ii=1{displaystyle alpha _{ij}={vec {v}}_{i}cdot {vec {v}}_{j}=alpha _{ji},alpha _{ii}=1}. The variables \u03b1ij,1\u2264i\u2192=(\u03b112,\u2026,\u03b11d,\u03b123,\u2026,\u03b1d\u22121,d)\u2208R(d2){displaystyle {vec {alpha }}=(alpha _{12},dotsc ,alpha _{1d},alpha _{23},dotsc ,alpha _{d-1,d})in mathbb {R} ^{binom {d}{2}}}. For a “congruent” integer multiexponent a\u2192=(a12,\u2026,a1d,a23,\u2026,ad\u22121,d)\u2208N0(d2),{displaystyle {vec {a}}=(a_{12},dotsc ,a_{1d},a_{23},dotsc ,a_{d-1,d})in mathbb {N} _{0}^{binom {d}{2}},} define \u03b1\u2192a\u2192=\u220f\u03b1ijaij{textstyle {vec {alpha }}^{vec {a}}=prod alpha _{ij}^{a_{ij}}}. Note that here N0{displaystyle mathbb {N} _{0}} = non-negative integers, or natural numbers beginning with 0. The notation \u03b1ji{displaystyle alpha _{ji}} for i}”\/> means the variable \u03b1ij{displaystyle alpha _{ij}}, similarly for the exponents aji{displaystyle a_{ji}}.Hence, the term \u2211m\u2260lalm{textstyle sum _{mneq l}a_{lm}} means the sum over all terms in a\u2192{displaystyle {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.References[edit]^ “Archimedes on Spheres and Cylinders”. Math Pages. 2015.^ a b Mazonka, Oleg (2012). “Solid Angle of Conical Surfaces, Polyhedral Cones, and Intersecting Spherical Caps”. arXiv:1205.1396 [math.MG].^ Hopf, Heinz (1940). “Selected Chapters of Geometry” (PDF). ETH Zurich: 1\u20132. Archived (PDF) from the original on 2018-09-21.^ “L’Huilier’s Theorem \u2013 from Wolfram MathWorld”. Mathworld.wolfram.com. 2015-10-19. Retrieved 2015-10-19.^ “Spherical Excess \u2013 from Wolfram MathWorld”. Mathworld.wolfram.com. 2015-10-19. Retrieved 2015-10-19.^ Eriksson, Folke (1990). “On the measure of solid angles”. Math. Mag. 63 (3): 184\u2013187. doi:10.2307\/2691141. JSTOR\u00a02691141.^ Van Oosterom, A; Strackee, J (1983). “The Solid Angle of a Plane Triangle”. IEEE Trans. Biomed. Eng. BME-30 (2): 125\u2013126. doi:10.1109\/TBME.1983.325207. PMID\u00a06832789. S2CID\u00a022669644.^ “Area of a Latitude-Longitude Rectangle”. The Math Forum @ Drexel. 2003.^ Jackson, FM (1993). “Polytopes in Euclidean n-space”. Bulletin of the Institute of Mathematics and Its Applications. 29 (11\/12): 172\u2013174.^ Aomoto, Kazuhiko (1977). “Analytic structure of Schl\u00e4fli function”. Nagoya Math. J. 68: 1\u201316. doi:10.1017\/s0027763000017839.^ Beck, M.; Robins, S.; Sam, S. V. (2010). “Positivity theorems for solid-angle polynomials”. Contributions to Algebra and Geometry. 51 (2): 493\u2013507. arXiv:0906.4031. Bibcode:2009arXiv0906.4031B.^ Ribando, Jason M. (2006). “Measuring Solid Angles Beyond Dimension Three”. Discrete & Computational Geometry. 36 (3): 479\u2013487. doi:10.1007\/s00454-006-1253-4.Further reading[edit]Jaffey, A. H. (1954). “Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables”. Rev. Sci. Instrum. 25 (4): 349\u2013354. Bibcode:1954RScI…25..349J. doi:10.1063\/1.1771061.Masket, A. Victor (1957). “Solid angle contour integrals, series, and tables”. Rev. Sci. Instrum. 28 (3): 191. Bibcode:1957RScI…28..191M. doi:10.1063\/1.1746479.Naito, Minoru (1957). “A method of calculating the solid angle subtended by a circular aperture”. J. Phys. Soc. Jpn. 12 (10): 1122\u20131129. Bibcode:1957JPSJ…12.1122N. doi:10.1143\/JPSJ.12.1122.Paxton, F. (1959). “Solid angle calculation for a circular disk”. Rev. Sci. Instrum. 30 (4): 254. Bibcode:1959RScI…30..254P. doi:10.1063\/1.1716590.Khadjavi, A. (1968). “Calculation of solid angle subtended by rectangular apertures”. J. Opt. Soc. Am. 58 (10): 1417\u20131418. doi:10.1364\/JOSA.58.001417.Gardner, R. P.; Carnesale, A. (1969). “The solid angle subtended at a point by a circular disk”. Nucl. Instrum. Methods. 73 (2): 228\u2013230. Bibcode:1969NucIM..73..228G. doi:10.1016\/0029-554X(69)90214-6.Gardner, R. P.; Verghese, K. (1971). “On the solid angle subtended by a circular disk”. Nucl. Instrum. Methods. 93 (1): 163\u2013167. Bibcode:1971NucIM..93..163G. doi:10.1016\/0029-554X(71)90155-8.Gotoh, H.; Yagi, H. (1971). “Solid angle subtended by a rectangular slit”. Nucl. Instrum. Methods. 96 (3): 485\u2013486. Bibcode:1971NucIM..96..485G. doi:10.1016\/0029-554X(71)90624-0.Cook, J. (1980). “Solid angle subtended by a two rectangles”. Nucl. Instrum. Methods. 178 (2\u20133): 561\u2013564. Bibcode:1980NucIM.178..561C. doi:10.1016\/0029-554X(80)90838-1.Asvestas, John S..; Englund, David C. (1994). “Computing the solid angle subtended by a planar figure”. Opt. Eng. 33 (12): 4055\u20134059. Bibcode:1994OptEn..33.4055A. doi:10.1117\/12.183402. Erratum ibid. vol 50 (2011) page 059801.Tryka, Stanislaw (1997). “Angular distribution of the solid angle at a point subtended by a circular disk”. Opt. Commun. 137 (4\u20136): 317\u2013333. Bibcode:1997OptCo.137..317T. doi:10.1016\/S0030-4018(96)00789-4.Prata, M. J. (2004). “Analytical calculation of the solid angle subtended by a circular disc detector at a point cosine source”. Nucl. Instrum. Methods Phys. Res. A. 521 (2\u20133): 576. arXiv:math-ph\/0305034. Bibcode:2004NIMPA.521..576P. doi:10.1016\/j.nima.2003.10.098.Timus, D. M.; Prata, M. J.; Kalla, S. L.; Abbas, M. I.; Oner, F.; Galiano, E. (2007). “Some further analytical results on the solid angle subtended at a point by a circular disk using elliptic integrals”. Nucl. Instrum. Methods Phys. Res. A. 580: 149\u2013152. Bibcode:2007NIMPA.580..149T. doi:10.1016\/j.nima.2007.05.055.External links[edit]Arthur P. Norton, A Star Atlas, Gall and Inglis, Edinburgh, 1969.M. G. Kendall, A Course in the Geometry of N Dimensions, No. 8 of Griffin’s Statistical Monographs & Courses, ed. M. G. Kendall, Charles Griffin & Co. Ltd, London, 1961Weisstein, Eric W. “Solid Angle”. MathWorld.Linear\/translational quantitiesAngular\/rotational quantitiesDimensions1LL2Dimensions1\u03b8\u03b82Ttime: tsabsement: Am sTtime: ts 1distance: d, position: r, s, x, displacementmarea: Am21angle: \u03b8, angular displacement: \u03b8radsolid angle: \u03a9rad2, srT\u22121frequency: fs\u22121, Hzspeed: v, velocity: vm s\u22121kinematic viscosity: \u03bd,specific angular momentum:\u00a0hm2 s\u22121T\u22121frequency: fs\u22121, Hzangular speed: \u03c9, angular velocity: \u03c9rad\u00a0s\u22121 T\u22122acceleration: am s\u22122T\u22122angular acceleration: \u03b1rad\u00a0s\u22122 T\u22123jerk: jm s\u22123T\u22123angular jerk: \u03b6rad\u00a0s\u22123 Mmass: mkgweighted position: M \u27e8x\u27e9 = \u2211 m x ML2moment of inertia:\u00a0Ikg\u00a0m2 MT\u22121momentum: p, impulse: Jkg\u00a0m\u00a0s\u22121, N saction: \ud835\udcae, actergy: \u2135kg\u00a0m2\u00a0s\u22121, J sML2T\u22121angular momentum: L, angular impulse: \u0394Lkg\u00a0m2\u00a0s\u22121action: \ud835\udcae, actergy: \u2135kg\u00a0m2\u00a0s\u22121, J\u00a0sMT\u22122force: F, weight: Fgkg m s\u22122, Nenergy: E, work: W, Lagrangian: Lkg m2 s\u22122, JML2T\u22122torque: \u03c4, moment: Mkg m2 s\u22122, N menergy: E, work: W, Lagrangian: Lkg m2 s\u22122, JMT\u22123yank: Ykg m s\u22123, N s\u22121power: Pkg m2 s\u22123,\u00a0WML2T\u22123rotatum: Pkg m2 s\u22123, N m s\u22121power: Pkg m2 s\u22123,\u00a0W"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/solid-angle-wikipedia\/#breadcrumbitem","name":"Solid angle – Wikipedia"}}]}]