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solid angle
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(Definition)
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A conical surface may contain a certain portion $\Omega$ of the space $\mathbb{R}^3$ . 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 $P$ in the vertex of the solid angle, then the magnitude of the solid angle is given by $$\Omega = \frac{A}{R^2}$$ which is 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 surface integral
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(1) |
where $r$ is the length of the position vector $\vec{r}$ for the points on the surface $a$ . The full solid angle, consisting of all points of $\mathbb{R}^3$ , has the magnitude $4\pi$ .
The SI unit of solid angle, analogous to the angle unit radian, is the 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 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 side line is equal to $2\pi(1\!-\cos\alpha)$ , i.e. $\displaystyle 4\pi\sin^2{\frac{\alpha}{2}}$ .
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 spanned by the vectors $\vec{r_i}$ is obtained from the equation
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(2) |
where the numerator of the right hand side is the triple scalar product of the vectors; the result is due to A. van Oosterom and J. Strackee 1983.
Example 3. Using (2), one can easily get the apical solid angle of a right pyramid 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 height of the pyramid.
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"solid angle" is owned by pahio.
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Cross-references: pyramid, side, base, square, triple scalar product, numerator, equation, vectors, tetrahedron, axis, right circular cone, planes, bounded, sphere, surface area, Proportion, radian, angle, Si, points, position vector, length, contained, radius, surface, contain, conical surface
There are 6 references to this entry.
This is version 18 of solid angle, born on 2005-07-25, modified 2009-03-22.
Object id is 7266, canonical name is SolidAngle.
Accessed 20145 times total.
Classification:
| AMS MSC: | 51M25 (Geometry :: Real and complex geometry :: Length, area and volume) | | | 15A72 (Linear and multilinear algebra; matrix theory :: Vector and tensor algebra, theory of invariants) |
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Pending Errata and Addenda
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