Game Development Reference
InDepth Information
The gradient operator in cylindrical coordinates is given by
∂
∂
∂
′
=
(
)
(
)
ˆ
∇
∇
rxyz
,,
+
∇
θ xyz
,,
+
z
.
(C.9)
∂
r
∂
θ z
∂
(
)
(
)
Using the definitions given in Equation (C.5) for
rxyz
and
,,
θ xyz
, we ob
,,
(
)
(
)
tain the following for the gradients
rxyz
,,
and
θ xyz
,,
.
∇
∇
∂ ∂ ∂
=++
∂
r
r
r
(
)
∇
rxyz
,,
i
j
k
x
∂
y
∂
z
x
y
=
i
+
j
x
2
+
y
2
x
2
+
y
2
=
i
cos
θ
+
j
sin
θ
=
r
ˆ
(C.10)
∂ ∂ ∂
=++
∂
θ
θ
θ
(
)
∇
θ xyz
,,
i
j
k
x
∂
y
∂
z
−
y
x
=
i
+
j
2
2
2
2
x
+
y
x
+
y
sin
θ
cos
θ
=−
i
+
j
r
r
1
ˆ
=
θ
(C.11)
r
Thus, the gradient operator can be written as
∂
1
ˆ
∂
∂
∇
′ =+
r
ˆ
ˆ
rr θ z
θ
+
z
.
(C.12)
∂
∂
∂
C.3 Spherical Coordinates
A point
P
is represented by the quantities
r
,
θ
, and
φ
in spherical coordinates. As
shown in Figure C.3,
r
is equal to the distance from the origin to the point
P
. The
angle
θ
is the azimuth representing the angle formed between the
x
axis and the
line connecting the projection of
P
onto the
x

y
plane to the origin (just as in cy
lindrical coordinates). The angle
φ
is called the
polar angle
and represents the
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