solid angleSolidAngle2005-07-25 18:51:002009-03-22 13:21:19Definitionposition vectorspace anglefull solid anglesteradiantrihedral angle% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}A conical surface may contain a certain portion $\Omega$ of the space $\mathbb{R}^3$.\, This portion is called \emph{solid angle} or \emph{space angle}.\, If the conical surface contains a portion $A$ of a spherical surface with radius $R$ and with \PMlinkname{centre}{Sphere} $P$ in the \PMlinkescapetext{vertex} of the solid angle, then the magnitude of the solid angle is given by
$$\Omega = \frac{A}{R^2}$$
which is \PMlinkescapetext{independent} 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 \PMlinkescapetext{surface integral}
\begin{align}
\Omega = -\int_a \vec{da}\cdot\nabla\frac{1}{r},
\end{align}
where $r$ is the length of the position vector $\vec{r}$ for the points on the surface $a$.
The \emph{full solid angle}, consisting of all points of $\mathbb{R}^3$, has the magnitude $4\pi$.
The SI \PMlinkescapetext{unit} of solid angle, analogous to the angle \PMlinkescapetext{unit} radian, is the \emph{steradian} ($= 1\;\mbox{sr}$). 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 \emph{trihedral angle}; cf. the example 2!\\
\textbf{Example 1.} The solid angle determined by a right circular cone with the angle $\alpha$ between its axis and \PMlinkescapetext{side line} is equal to $2\pi(1\!-\cos\alpha)$, i.e. $\displaystyle 4\pi\sin^2{\frac{\alpha}{2}}$.\\
\textbf{Example 2.} Let\, $\vec{r_1},\,\vec{r_2},\,\vec{r_3}$ be the position vectors of three points in $\mathbb{R}^3$ and $r_1,\,r_2,\,r_3$ their lengths. Then the solid angle $\Omega$ of the tetrahedron \PMlinkescapetext{spanned} by the vectors $\vec{r_i}$ is obtained from the equation
\begin{align}
\tan{\frac{\Omega}{2}} = \frac{(\vec{r_1}\vec{r_2}\vec{r_3})}
{\vec{r_1}\!\cdot\!\vec{r_2}r_3+\vec{r_2}\!\cdot\!\vec{r_3}r_1+
\vec{r_3}\!\cdot\!\vec{r_1}r_2+r_1r_2r_3},
\end{align}
where the numerator of the \PMlinkescapetext{right hand side} is the triple scalar product of the vectors; the result is due to A. van Oosterom and J. Strackee 1983.\\
\textbf{Example 3.} Using (2), one can easily get the \PMlinkname{apical}{ConeInMathbbR3} solid angle of a \PMlinkname{right pyramid}{ConeInMathbbR3} with square base:
$$\Omega = 4\arctan{\frac{a^2}{2h\sqrt{2a^2+4h^2}}} = 4\arcsin\frac{a^2}{a^2+4h^2}$$
Here $a$ is the side of the base square and $h$ is the \PMlinkname{height}{ConeInMathbbR3} of the pyramid.