A solid angle
(measured in steradians "sr") is assigned to the cone generated by half a straight line
originating at the center of a sphere of unit radius with one point of that line moving
in a closed loop at the surface of the unit sphere.
The measure of such a solid angle is simply the spherical surface area enclosed by
the aforementioned loop at the surface of the unit sphere.
Just like a planar angle, a solid angle can be oriented
(i.e., assigned a sign) according to the direction in which the loop is traveled.
The usual convention is to count a solid angle positively if the loop is traveled
clounterclockwise when seen from the outside of the sphere or, equivalently,
clockwise seen from the origin at the center of the sphere.
You may memorize this by recalling that the south face of a loop
is seen at a positive solid angle (using the usual
convention to define the "north" and "south" side of an oriented loop).
That loop may cross itself many times: The spherical area so
enclosed is tallied algebraically as in the planar case we
describe elsewhere.
Just like planar angles are defined modulo
2p,
solid angles are defined up to a multiple of 4p,
because such is the entire surface area of a unit sphere
A "spat" is the solid angle subtended by the whole sphere
(4p).
Indeed, consider that a solid angle A changes to -A
when you reverse the direction of its defining loop.
However, you could also consider that the solid angle has become
(4p-A)
because the new orientation of the loop makes it enclose (as its "south side")
whatever part of the sphere was not previously enclosed by the loop as originally
oriented.
You may use this argument to convince yourself that multiples of
4p
are as irrelevant to solid angles as multiples of
2p
are irrelevant to planar angles.
The steradian is not the only unit of solid angle.
Astronomers routinely express solid angles in square degrees,
they also use square minutes or square seconds
for tiny solid angles.
Indeed, if a "rectangular" patch of sky is so small that the curvature of the celestial
sphere is negligible, then its surface is almost flat and it has an area very nearly equal
to the product of its angular width by its angular height (technically,
those concepts of "width" and "height" become precise only in the context of that flat
approximation). The result is in steradians if those angles are given
in radians. On the other hand,
if such angles are given in degrees, then the result is, by
definition, obtained in "square degrees". The square degree
is thus just a practical unit of solid angle which could be used to measure solid
angles of any size, although the aforementioned "small angle"
computation is only valid for very tiny rectangular patches of the sphere.
1 square degree =
( p / 180 ) 2 =
0.0003046... sr
A square minute is 602 = 3600 times smaller
than that.
A square second is 12960000 times smaller than a
square degree; it's roughly 2.35 10 -11 sr.
The whole celestial sphere (twice the sky) corresponds to a solid angle of
1 spat = 4p [sr] =
41252.96... square degrees
Band (between two parallel lines) :
The dihedral angle (q)
formed by two half-planes is proportional to the solid angle
(W) between them
for an observer located at any point O on their axis of interesection.
The coefficient of proportionality is simply obtained from
any special case...
In particular, if q = p/2,
the faces are perpendicular and the area of the spherical lune
between them is a quarter of the whole sphere
(W = p). Therefore, in general:
W = 2 q
This can be interpreted as the solid angle between two parallel lines
whose largest angular separation is seen to be
q (by an observer at point O).
Disk :
The part of a solid which lays between two parallel planes that intersect it
is called a frustum.
The height (h) of the frustum is the
distance between those planes.
For a sphere of radius R,
the lateral (spherical) surface of a frustum
has an area equal to 2pRh
(remarkably, for a sphere, this
area doesn't depend on the position of the cutting planes;
it's just a function of the distance between them).
With h = 2R,
we retrieve the surface area of the entire sphere, namely
4pR2.
In particular, with R = 1, a spherical cap
of angular radius q has a height
h = 1-cos q and a
surface area equal to the solid angle seen from O, namely:
W =
2p ( 1 - cos q )
=
4p sin2 (q/2)
Note that the latter expression is required
for accurate floating-point computations with small values of
q.
You may check that a solid angle of 2p
is obtained for the entire sky (a hemisphere, q=p/2)
and 4p for the whole sphere
(q=p).
The mean angular diameter of the full moon is
2q = 0.52° (it varies with time around
that average, by about 0.009°).
This translates into a solid angle of
0.0000647 sr, which means that the whole night sky
covers a solid angle roughly
one hundred thousand times greater than the full moon.
Symmetrical Rectangular Patch :
The solid angle of a rectangular field of view (symmetrical with respect to
the axis of observation) of angular width
2a and
height 2b is:
W =
4 arcsin ( sin(a) sin(b) )
For small angles, the patch is nearly flat and the surface of the rectangle is
nearly the product of its angular width by its angular height
W » (2a)(2b).
The above is, of course, the origin of the units of solid angle
commonly used by astronomers: square degree, square minute, etc.
The conversion factors between those units are always obtained in
the limit of very small angles.
For example, a square degree, expressed in steradians is simply
(p/180)2.
The above exact formula shows that a square patch of sky 1° by 1°
is only about 0.9999746 of a square degree!
The larger the angular size, the greater the discrepancy...
Note that a symmetrical square field, 1 radian on
a side, is about 0.9193954 of a "square radian"
(which is just an unused—and confusing—alternative name for a
steradian ).
For such a patch, the true solid angle exceeds the naive value by less
than 9%. For the whole sky (hemisphere) however,
ignoring curvature like that would result in a discrepancy of 57%
(the true solid angle is 2p,
not p2 ).
Triangles :
W =
j -
Arctg ( cos q tg j )
Regular Polygons :
At the apex O of a regular n-gonal pyramid
(i.e., a straight pyramid whose
base is a regular polygon with n sides) the solid angle subtended by the
base consists of 2n triangular solid angles of the type just discussed,
where j is
p/n and
q is the angular radius of the circle
circumscribed to the base. This adds up to:
W
=
2p -
2n Arctg ( cos q tg p/n )
For large values of n, we retrieve the disk formula:
W
=
2p (1 - cos q)
