Graphics Reference
In-Depth Information
Units: Just as angles are measured in radians, solid angles are measured in
steradians, abbreviated “sr.” The entire unit sphere has a solid angle measure of
4
π
steradians.
26.6.4 Computations with Solid Angles
Let's now measure a few simple solid angles (see Figure 26.21). Following the
standard from graphics rather than mathematics, we will treat the y -axis as point-
ing up, the x -axis as pointing to the right, and the z -axis as pointing toward us.
Thus, the longitude is
atan2
( y , z ) and the latitude is arcsin( y ) .(The colatitude,
which is often denoted
φ
in spherical polar coordinates, is arccos( y ) . The other
polar coordinate,
θ
, is what we've called the longitude.)
•f Ω is all of S 2 , then the measure of Ω is 4
π
(the area of a unit sphere).
• Any hemisphere has measure 2
.
• The “stripe” between y = y 0 and y = y 1 has area 2
π
π
y 1
y 0
. This follows
from the theorem below, as do the next two examples.
• The latitude-longitude rectangle between latitudes
λ 0 and
λ 1 and longi-
tudes
θ 0 and
θ 1 has solid angle
θ 1 −θ 0 ·
sin
λ 1
sin
λ 0
(where latitude
2 at the North Pole). (When the
longitudes are on opposite sides of the international dateline, this rectangle
is a very long stripe wrapping around the nondateline part of the globe.)
• A “disk” consisting of all points whose spherical distance from a point P
is less than r (where r
goes from
−π/
2 at the South Pole to
π/
) has solid angle measure 2
π
( 1
cos( r )) .
If a regular solid of n sides (cube ( n = 6 ) , tetrahedron ( n = 4 ) , octahedron
( n = 8 ) , dodecahedron ( n = 12 ) , icosahedron ( n = 20 )) is inscribed
in the unit sphere, the projection of one of its faces onto the sphere (see
Figure 26.21 (f)) has solid angle measure
4 n
, because the total projected
area is 4
π
, and by symmetry, each face has the same projected area.
All of the results above are consequences of the sphere-to-cylinder projec-
tion theorem: If C is a cylinder of radius 1 and height 2, circumscribed about the
sphere S of radius 1, then the horizontal radial projection map,
x
x 2 + z 2 , y ,
,
z
x 2 + z 2
p : C
S :( x , y , z )
(26.22)
is area-preserving. (The proof is a simple calculus computation—see Exer-
cise 26.1). Figure 26.22 shows this: The area of a country on the surface of the
globe is the same as the area on the plate carrée projection shown (although many
other characteristics of shape are grossly distorted, as shown for Greenland [in
green]).
As an example of another use of this theorem, let's let Ω denote the north-
ern hemisphere y
0 of the unit sphere, and integrate the function y over this
hemisphere.
That is to say, we seek to evaluate
B =
ydA .
(26.23)
( x , y , z ) Ω
 
 
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