Solid angle
Solid angle | ||
---|---|---|
SI unit steradian | | |
Other units | Square degree, spat (angular unit) | |
In SI base units | m2/m2 | |
Conserved? | No | |
Derivations from other quantities | ||
Dimension |
In geometry, a solid angle (symbol: Ω) 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, . 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 (and therefore apparent size). This is evident during a solar eclipse.
Definition and properties
An object's solid angle in
where is the spherical surface area and 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.
The solid angle of a sphere measured from any point in its interior is 4
In
where θ is the colatitude (angle from the North Pole) and φ is the longitude.
The solid angle for an arbitrary
where is the unit vector corresponding to , the
Thus one can approximate the solid angle subtended by a small facet having flat surface area dS, orientation , and distance r from the viewer as:
where the
Practical applications
- Defining luminous intensity and luminance, and the correspondent radiometric quantities radiant intensity and radiance
- Calculating 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
- Calculating the electric field and magnetic field 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
- The solid angle of the acceptance cone of the optical fiber
Solid angles for common objects
Cone, spherical cap, hemisphere
The solid angle of a
For small θ such that cos θ ≈ 1 − θ2/2 this reduces to πθ2, the area of a circle.
The above is found by computing the following
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 as
Hence for a unit sphere the solid angle of the spherical cap is given as
When θ = π/2, the spherical cap becomes a hemisphere having a solid angle 2π.
The solid angle of the complement of the cone is
This is also the solid angle of the part of the celestial sphere that an astronomical observer positioned at latitude θ 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 γ from the cone's axis and passing through the cone's apex can be calculated by the formula[2]
For example, if γ = −θ, then the formula reduces to the spherical cap formula above: the first term becomes π, and the second π cos θ.
Tetrahedron
Let OABC be the vertices of a tetrahedron 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 θa to be the angle BOC and define θb, θc correspondingly. Let be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define , correspondingly. The solid angle Ω subtended by the triangular surface ABC is given by
This follows from the theory of
where 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 θa, θb, θc is given by L'Huilier's theorem[4][5] as
where
Another interesting formula involves expressing the vertices as vectors in 3 dimensional space. Let 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 Ω subtended by the triangular surface ABC is:[6][7]
where
denotes the scalar triple product of the three vectors and denotes the
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 π.
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
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 sphere) are known, then the above equation can be manipulated to give
The solid angle of a right n-gonal pyramid, where the pyramid base is a regular n-sided polygon of circumradius r, with a pyramid height h is
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]
where parentheses (* *) is a
Latitude-longitude rectangle
The solid angle of a latitude-longitude rectangle on a globe is
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
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, , and the distance from the observer to the object, :
By inputting the appropriate average values for the
Solid angles in arbitrary dimensions
The solid angle subtended by the complete (d − 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
This gives the expected results of 4π steradians for the 3D sphere bounded by a surface of area 4πr2 and 2π radians for the 2D circle bounded by a circumference of length 2πr. It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D "sphere" is the interval [−r, 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:
References
- ^ "Archimedes on Spheres and Cylinders". Math Pages. 2015.
- ^ ].
- ^ Hopf, Heinz (1940). "Selected Chapters of Geometry" (PDF). ETH Zurich: 1–2. Archived (PDF) from the original on 2018-09-21.
- ^ "L'Huilier's Theorem – from Wolfram MathWorld". Mathworld.wolfram.com. 2015-10-19. Retrieved 2015-10-19.
- ^ "Spherical Excess – from Wolfram MathWorld". Mathworld.wolfram.com. 2015-10-19. Retrieved 2015-10-19.
- JSTOR 2691141.
- S2CID 22669644.
- ^ "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): 172–174.
- .
- Bibcode:2009arXiv0906.4031B.
- .
Further reading
- 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): 349–354. .
- Masket, A. Victor (1957). "Solid angle contour integrals, series, and tables". Rev. Sci. Instrum. 28 (3): 191. .
- Naito, Minoru (1957). "A method of calculating the solid angle subtended by a circular aperture". J. Phys. Soc. Jpn. 12 (10): 1122–1129. .
- Paxton, F. (1959). "Solid angle calculation for a circular disk". Rev. Sci. Instrum. 30 (4): 254. .
- Khadjavi, A. (1968). "Calculation of solid angle subtended by rectangular apertures". J. Opt. Soc. Am. 58 (10): 1417–1418. .
- Gardner, R. P.; Carnesale, A. (1969). "The solid angle subtended at a point by a circular disk". Nucl. Instrum. Methods. 73 (2): 228–230. .
- Gardner, R. P.; Verghese, K. (1971). "On the solid angle subtended by a circular disk". Nucl. Instrum. Methods. 93 (1): 163–167. .
- Gotoh, H.; Yagi, H. (1971). "Solid angle subtended by a rectangular slit". Nucl. Instrum. Methods. 96 (3): 485–486. .
- Cook, J. (1980). "Solid angle subtended by a two rectangles". Nucl. Instrum. Methods. 178 (2–3): 561–564. .
- Asvestas, John S..; Englund, David C. (1994). "Computing the solid angle subtended by a planar figure". Opt. Eng. 33 (12): 4055–4059. . 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 (4–6): 317–333. .
- 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 (2–3): 576. S2CID 15266291.
- 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: 149–152. .
External links
- 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, 1961
- Weisstein, Eric W. "Solid Angle". MathWorld.