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In-Depth Information
15
16
sin
2
Y
2
,−
2
=
θ
cos 2
φ
π
4π
15
4
Y
1
,−
1
=
−
sinθ cos φ
Y
2
,−
1
=
−
sinθ cos θ cos φ
π
4π
4π
5
16
π
(
3cos
2
Y
0
,
0
=
Y
1
,
0
=
cos θ
Y
2
,
0
=
θ
−
1
)
4π
15
4
Y
1
,
1
=
−
sinθ sinφ
Y
2
,
1
=
−
sinθ cos θ sinφ
π
15
16π
sin
2
Y
2
,
2
=
θ sin2φ
Tabl e 7. 1
Real spherical harmonics for
l
=
0
,
1
,
2.
where
K
l
is a normalizing factor, typically defined by
(
2
l
+
1
)(
l
−|
m
|
)
!
K
k
=
4
π
(
l
+
|
m
|
)
!
although there are other definitions.
4
The
P
l
are the
associated Legendre poly-
nomials
. There is no direct formula for the coefficients of these polynomials.
They can be defined in several ways, such as,
(
x
)
m
2
l
l
!
(
2
d
l
+
m
dx
l
+
m
(
)=
(
−
1
)
P
l
x
2
m
/
x
2
l
(
x
1
−
)
−
1
)
−
l
≤
m
≤
l
;
however, the recurrence formula
xP
l
P
l
−
1
(
)=
(
2
l
+
1
)
(
x
)
−
(
l
+
m
)
x
)
P
l
+
1
(
x
(
−
+
)
l
m
1
isoftenusedinpractice.
When applied to real-valued functions, a “real form” of the basis functions
Y
l
can be used instead. These
real spherical harmonics
are
√
2Re
Y
l
)=
√
2
K
l
⎧
⎨
P
l
(
cos
(
m
φ
)
(
cos
θ
)
,
m
>
0
,
√
2Im
Y
l
)=
√
2
K
l
sin
P
−
m
l
Y
l
,
m
=
(
(
−
m
φ
)
(
cos
θ
)
,
m
<
0
,
⎩
Y
l
K
l
P
l
(
=
cos
θ
)
,
m
=
0
,
and are distinguished from the complex versions by putting both indices in the
subscript. Table 7.1 contains the first few real SH basis functions.
Spherical harmonic expansions of environment maps.
An expression
of the form Equation (7.2) is known as a
spherical harmonic expansion
. In prac-
tice, the series is usually limited to a finite number of terms, in which case the
4
Sometimes the 4π term is omitted, and an extra sign term
(
−
1
)
m
is sometimes added.
