Graphics Reference
In-Depth Information
Let's apply the ideas of the previous section to the longitude-colatitude parame-
terization of
S
2
, namely,
S
2
:(
u
,
v
)
Y
:[
0, 1
]
×
[
0, 1
]
→
→
(cos(
2
π
u
)sin(
π
v
)
,
cos(
π
v
)
,
sin(
2
π
u
)sin(
v
))
.
(30.35)
π
(
u
,
)
Area
=
A
We'll start with the uniform probability density
p
on
[
0, 1
]
[
0, 1
]
,or
p
(
u
,
v
)=
1
for all
u
,
v
. For notational convenience, we'll write
(
x
,
y
,
z
)=
Y
(
u
,
v
)
.
We have the intuitive sense that if we pick points uniformly randomly in the
unit square and use
f
to convert them to points on the unit sphere, they'll be
clustered near the poles. (If you doubt this, you should write a little program to
verify it.) This means that the induced probability density on
T
will not be uni-
form.
The preceding section shows that
×
u
(
x
,
y
,
z
)
1
Area
=
p
Y
(
x
,
y
,
z
)=
p
(
u
,
v
)
(30.36)
2
|
Y
(
u
,
v
)
|
A
?
2
π
si
n
(
π
)
1
=
.
(30.37)
Y
(
u
,
v
)
|
|
But the change-of-area factor (see Figure 30.9) for
Y
(which is slightly messy
to compute) turns out to be
Figure 30.9: A small area A
in the domain of the spherical
parameterization gets multiplied
by
2
π
Y
(
u
,
v
)
2
|
|
=
2
π
|
sin(
π
v
)
|
(30.38)
2
1
2
sin(
π
v
)
.
=
2
π
−
cos(
π
v
)
2
(30.39)
2
1
=
2
π
−
y
2
.
(30.40)
And hence,
1
2
1
p
Y
(
x
,
y
,
z
)=
y
2
.
(30.41)
2
π
−
Thus, the probability of sampling a point in a sm
all dis
k of area
A
centered on the
sphere point
(
x
,
y
,
z
)
is approximately
A
2
1
−
y
2
)
.
/
(
2
π
If we start with the uniformly distributed random variable
U
on
[
0, 1
]
×
[
0, 1
]
and
want a uniformly distributed variable
V
on, say,
[
0, 2
π
]
×
[
0, 1
]
, it seems obvious
to define
V
(
a
,
b
)=
U
(
2
π
,
b
)
. The density for
U
is the function
p
U
:[
0, 1
]
×
[
0, 1
]
→
R
:(
x
,
y
)
→
1.
(30.42)
That makes the density for
V
be the function
1
2
p
V
:[
0, 2
π
]
×
[
0, 1
]
→
R
:(
x
,
y
)
→
.
(30.43)
π
(Quick proof:
V
is evidently uniformly distributed, and hence its pdf is a constant.
The integral of that constant over the domain must be 1, so the constant is
1
2
.)
π
