Solid Angle Article Index for
Solid 		Website Links For
Solid 		 

Information About

Solid Angle
  CATEGORIES ABOUT SOLID ANGLE
   
angle
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



If the proportionality constant is chosen to be 1, the units of solid angle will be the SI Steradian (abbreviated ''sr''). Thus the solid angle of a sphere measured at its center is 4 π sr, and the solid angle subtended at the center of a cube by one of its sides is one-sixth of that, or 2π/3 sr. Solid angles can also be measured (for ''k''=(180/π)&2) in Square Degree s or (for ''k''=1/4π) in fractions of the sphere (i.e., ''fractional area'').

One way to determine the fractional area subtended by a spherical surface is to divide the area of that surface by the entire surface area of the sphere. The fractional area can then be converted to steradian or square degree measurements by the following formulae:

#To obtain the solid angle in steradians, multiply the fractional area by 4π.
#To obtain the solid angle in square degrees, multiply the fractional area by 4π × (180/π)&2, which is equal to 129600/π.


PRACTICAL APPLICATIONS




SOLID ANGLES FOR COMMON OBJECTS


  • An efficient algorithm for calculating the solid angle Ω 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):



an \left( rac{1}{2} \Omega ight)
=
rac{ {\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}}

,

where:
: R1R2R3" class="copylinks" target="_blank">{Link without Title} denotes the Determinant of the matrix that results when writing the vectors together in a row, e.g. Mij=Rj(i);
: ''R''i denotes the distance of point i from the origin and Ri is the vector representation of point i;
: Ri·Rj denotes the Scalar Product .


  • The solid angle of a Cone with apex angle a is 2 \pi \left (1 - \cos {a \over 2} ight).


  • The solid angle of a four-sided right regular Pyramid with apex angle a (measured to the faces of the pyramid) is 4 \arccos \left (-\sin^2 {a \over 2} ight) - 2 \pi.


  • The Sun and Moon are both seen from Earth at a ''fractional area'' of 0.001% of the celestial hemisphere.



SOLID ANGLE IN ARBITRARY DIMENSION


The volume of the unit sphere can be defined in any dimension. One often needs this solid angle factor in calculations with spherical symmetry.



\Omega_{d}
=
rac{2\pi^{d/2}}{\Gamma \left ( rac{d}{2} ight )}



Where Γ is the Gamma Function .