立体角Solid Angle
Solid Angle
The solid angle
subtended by a surface is defined as the surface area
of a unit sphere covered by the surface's projection onto the sphere. This can be written as
where
is a unit vector from the origin,
(1)
is the differential area of a surface patch, and is
the distance from the origin to the patch. Written in spherical coordinates
with the colatitude
(polar angle) and for the longitude (azimuth), this becomes
(2)
Solid angle is measured in steradians , and the solid angle corresponding to all of space being
subtended is
steradians .
To see how the solid angle of simple geometric shapes can be computed explicitly, consider
the solid angle
subtended by one face of a cube of side length
centered at the
origin. Since the cube is symmetrical and has six sides, one side obviously subtends
steradians. To compute this explicitly, rewrite (1) in Cartesian coordinates
using
and
(3)
(4)
(5)
(6)
and has sides parallel the -
Considering the top face of the cube, which is located at
and -axes,
(7)
(8)
as expected.
Similarly, consider a tetrahedron with side lengths
with origin at the centroid, base at
(where
is the centroid), and bottom vertices at
and
, where
(9)
(10)
(11)
Then
runs from
to
, and for the half of the base in the positive half-plane
, can be
taken to run from 0 to
i.e.,
, as expected.
, giving
(12)
(13)