solid angle

A conical surface may contain a certain portion $\mathrm{\Omega }$ of the space ${ℝ}^3$.  This portion is called solid angle or space angle.  If the conical surface contains a portion $A$ of a spherical surface with radius $R$ and with centre (http://planetmath.org/Sphere) $P$ in the of the solid angle, then the magnitude of the solid angle is given by

$$\mathrm{\Omega }=\frac{A}{R^2}$$

which is on the radius $R$. The spherical surface can be replaced by any surface $a$, through which all the half-lines originating from $P$ and being contained in the solid angle go.  Then the solid angle may be computed from the

$\mathrm{\Omega }=-{\displaystyle {\int }_a}\overrightarrow{da}\cdot \nabla {\displaystyle \frac{1}{r}},$ (1)

where $r$ is the length of the position vector $\overrightarrow{r}$ for the points on the surface $a$. The full solid angle, consisting of all points of ${ℝ}^3$, has the magnitude $4\pi$.

The SI of solid angle, analogous to the angle radian, is the steradian ($=1\text{sr}$).  The steradian takes a proportion $\frac{1}{4\pi }$, or approximately 7.957747 %, of the surface area of a sphere.

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 $\alpha$ between its axis and is equal to $2\pi (1-\cos \alpha )$, i.e. $4\pi {\sin }^2{\displaystyle \frac{\alpha }{2}}$.

Example 2.  Let  ${\overrightarrow{r}}_1,{\overrightarrow{r}}_2,{\overrightarrow{r}}_3$ be the position vectors of three points in ${ℝ}^3$ and $r_1,r_2,r_3$ their lengths.  Then the solid angle $\mathrm{\Omega }$ of the tetrahedron by the vectors $\overrightarrow{r_i}$ is obtained from the equation

$\tan {\displaystyle \frac{\mathrm{\Omega }}{2}}={\displaystyle \frac{{\overrightarrow{r}}_1\times {\overrightarrow{r}}_2\cdot {\overrightarrow{r}}_3}{({\overrightarrow{r}}_1\cdot {\overrightarrow{r}}_2)r_3+({\overrightarrow{r}}_2\cdot {\overrightarrow{r}}_3)r_1+({\overrightarrow{r}}_3\cdot {\overrightarrow{r}}_1)r_2+r_1{r}_2{r}_3}},$ (2)

where the numerator of the is the triple scalar product of the vectors.  This equation is expressed simplier using the unit vectors ${\overrightarrow{u}}_i$ corresponding ${\overrightarrow{r}}_i$:

$$\tan \frac{\mathrm{\Omega }}{2}=\frac{{\overrightarrow{u}}_1\times {\overrightarrow{u}}_2\cdot {\overrightarrow{u}}_3}{1+{\overrightarrow{u}}_2\cdot {\overrightarrow{u}}_3+{\overrightarrow{u}}_3\cdot {\overrightarrow{u}}_1+{\overrightarrow{u}}_1\cdot {\overrightarrow{u}}_2}$$

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:

$$\mathrm{\Omega }=\mathrm{\hspace{0.33em}4}\arctan \frac{a^2}{2h\sqrt{2a^2+4h^2}}=\mathrm{\hspace{0.33em}4}\arcsin \frac{a^2}{a^2+4h^2}$$

Here $a$ is the side of the base square and $h$ is the height (http://planetmath.org/ConeInMathbbR3) of the pyramid.  Cf. the solid angle of rectangular pyramid.

Title	solid angle
Canonical name	SolidAngle
Date of creation	2014-03-12 21:18:06
Last modified on	2014-03-12 21:18:06
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	26
Author	pahio (2872)
Entry type	Definition
Classification	msc 15A72
Classification	msc 51M25
Related topic	ConvexAngle
Related topic	Radian
Related topic	AreaOfASphericalTriangle
Related topic	DihedralAngle
Defines	space angle
Defines	full solid angle
Defines	steradian
Defines	trihedral angle