Solid angle
A solid angle is the three dimensional analog of the ordinary angle. Instead of two lines meeting at a vertex, though, one needs a three dimensional figure that meets at a point. Simple examples of objects that do this are a cone or a pyramid. The SI unit of solid angle is the steradian (symbol sr), which is equal to radian2. Solid Angle can also be measured in degrees2.
To find the solid angle that an object subtends at a point, imagine a sphere centered at the point. Now, divide the surface area of the part of the sphere that is contained within the outline of the object by the total area of the sphere to obtain the fractional area.
- To obtain the solid angle in steradians or radians squared, multiply the fractional area by 4π.
- To obtain the solid angle in degrees squared, multiply the fractional area by 4 x 1802/π which is equal to 129600/π.
By analogy with the two dimensional case--
- To get an angle, imagine two lines passing through the center of a unit circle. The length of the arc between the lines on the unit circle is the angle, in radians.
- To get a solid angle, imagine three or more planes passing through the center of a unit sphere. The area of the surface between the planes on the unit sphere is the solid angle, in steradians. The angle between two planes is termed dihedral, between three trihedral, between any number more than three polyhedral. A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs on a sphere, and is measured by the angle between the planes containing the arcs and the centre of the sphere.
Solid angle is useful for...
- defining luminosity
- calculating spherical excess E of a spherical triangle
- the calculation of potentials by using the Boundary Element Method (BEM)
An efficient algorithm for calculating the solid angle subtended by a triangle with vertices R1, R2 and R3, as seen from the origin has been given by Oosterom and Strackee (IEEE Trans. Biom. Eng., Vol BME-30, No 2, 1983):
![\tan \left( \frac{1}{2} \Omega \right) = \frac{[ {\mathbf R}_{1}{\mathbf R}_{2}{\mathbf R}_{3}]}{ R_{1}R_{2}R_{3} + ( {\mathbf R}_{1} \cdot {\mathbf R}_{2})R_{3} + ( {\mathbf R}_{1} \cdot {\mathbf R}_{3})R_{2} + ( {\mathbf R}_{2} \cdot {\mathbf R}_{3})R_{1}}](/img/math/1943f0d78dbb5fc2c86a441e28b97f95.png)
.
The Sun and Moon are both seen from Earth at a fractional area of 0.001% of the celestial hemisphere.