Tuesday, March 24, 2009

Solid Angle

Solid Angle

The solid angle Omega subtended by a surface S is defined as the surface area Omega of a unit sphere covered by the surface's projection onto the sphere. This can be written as

 Omega=intint_S(n^^・da)/(r^2),

where n^^ is a unit vector from the origin, da is the differential area of a surface patch, and r is the distance from the origin to the patch. Written in spherical coordinates with phi the colatitude (polar angle) and theta for the longitude (azimuth), this becomes

 Omega=intint_Ssinphidthetadphi.

Solid angle is measured in steradians, and the solid angle corresponding to all of space being subtended is 4pi steradians.

SolidAngleCube

To see how the solid angle of simple geometric shapes can be computed explicitly, consider the solid angle Omega_(cube side) subtended by one face of a cube of side length 2a centered at the origin. Since the cube is symmetrical and has six sides, one side obviously subtends 4pi/6= steradians. To compute this explicitly, rewrite (1) in Cartesian coordinates using

n^^・da=cosphidxdy
r^2=x^2+y^2+z^2

and

cosphi=z/r
=z/(sqrt(x^2+y^2+z^2)).

Considering the top face of the cube, which is located at z= and has sides parallel the x- and y-axes,

Omega_(cube side)=int_(-a)^aint_(-a)^a(adxdy)/((x^2+y^2+a^2)^(3/2))
=2/3pi,

as expected.

SolidAngleTetrahedron

Similarly, consider a tetrahedron with side lengths a with origin at the centroid, base at z= (where r is the centroid), and bottom vertices at (-d,+/-a/2,-r) and (x_0,0,-r), where

x_0=1/3sqrt(3)a
r=1/(12)sqrt(6)a
d=1/6sqrt(3)a.

Then x runs from -d to x_0, and for the half of the base in the positive y half-plane, y can be taken to run from 0 to a/2-(a/2)/(d+x_0)(x+d)=, giving

Omega=2int_(-d)^(x_0)int_0^(1/3-x/sqrt(3))(rdydx)/((x^2+y^2+r^2)^(3/2))
=pi,

i.e., 4pi/4, as expected.

NOTE: obtained from http://mathworld.wolfram.com/SolidAngle.html

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