A conical surface may contain a certain portion of the space . This portion is called solid angle or space angle. If the conical surface contains a portion of a spherical surface with radius and with centre (http://planetmath.org/Sphere) in the of the solid angle, then the magnitude of the solid angle is given by
which is on the radius . The spherical surface can be replaced by any surface , through which all the half-lines originating from and being contained in the solid angle go. Then the solid angle may be computed from the
where is the length of the position vector for the points on the surface . The full solid angle, consisting of all points of , has the magnitude .
If the solid angle is bounded by three planes having exactly one common point, it may be called a trihedral angle; cf. the example 2!
Example 1. The solid angle determined by a right circular cone with the angle between its axis and is equal to , i.e. .
Example 2. Let be the position vectors of three points in and their lengths. Then the solid angle of the tetrahedron by the vectors is obtained from the equation
The result (2) is due to van Oosterom and Strackee 1983.
Example 3. Using (2), one can easily get the apical (http://planetmath.org/ConeInMathbbR3) solid angle of a right pyramid (http://planetmath.org/ConeInMathbbR3) with square base:
- 1 A. van Oosterom, J. Strackee: A solid angle of a plane triangle. – IEEE Trans. Biomed. Eng. 30:2 (1983); 125–126.
- 2 M. S. Gossman, A. J. Pahikkala, M. B. Rising, P. H. McGinley: Providing Solid Angle Formalism for Skyshine Calculations. – Journal of Applied Clinical Medical Physics. 11:4 (2010); 278–282.
|Date of creation||2014-03-12 21:18:06|
|Last modified on||2014-03-12 21:18:06|
|Last modified by||pahio (2872)|
|Defines||full solid angle|