Graphics Reference
In-Depth Information
2
(
y
2
-
y
1
)
y
2
4
2
y
1
(a)
(b)
(c)
(
2
-
1
)(cos(
2
) - cos(
1
))
P
1
r
2
2
(1 - cos(
r
))
2
/3
(d)
(e)
(f)
Figure 26.21: Various solid angles on the unit sphere.
(
x
,
y
,
z
):
x
2
+
z
2
=
1, 0
Consider the upper half-cylinder
H
=
,
which projects to
Ω
under the axial projection map
p
. We can perform a change
of variables in the integral, and express
B
as
B
=
(
x
,
y
,
z
)
∈
H
{
≤
y
≤
1
}
Jp
(
x
,
y
,
z
)
dA
,
y
|
|
(26.24)
where
(
x
,
y
,
z
)=
p
(
x
,
y
,
z
)
, and
dA
is area on
H
, and
is the Jacobian for
the change of variables (i.e., it represents how areas at
(
x
,
y
,
z
)
are stretched
or contracted to become areas at
(
x
,
y
,
z
))
. The theorem that
p
is area-preserving
means that
|
Jp
|
|
Jp
|
=
1, so the integral becomes
B
=
ydA
.
(26.25)
(
x
,
y
,
z
)
∈
H
Figure 26.22: Horizontal radial
projection from the sphere to
the surrounding cylinder is area-
preserving.
Since in the formula for
p
,
y
does not change, we have
y
=
y
, so this becomes
B
=
(
x
,
y
,
z
)
y
dA
.
(26.26)
∈
H
times the integral of
y
from 0 to 1. That
By circular symmetry, this is just 2
π
2, so
B
=
2
·
π
=
π
integral is 1
/
2
.
If instead we wanted to know the
average
of
y
over the upper hemisphere, we'd
need to divide its integral (
π
) by the area of the hemisphere (2
π
). The average is
1
thus
2
. This value comes up often, although it's usually in a slightly generalized
form: We have a hemisphere defined by
v
·
n
≥
0, and we want to know the
average value of
n
over this hemisphere. (Our instance is the special case
where
n
=
010
T
.) We'll state this as a principle:
v
·
