Game Development Reference
InDepth Information
The gradient operator
has the following form in Cartesian coordinates.
∇
∂ ∂ ∂
≡++
∂
∇
i
j
k
(C.2)
x
∂
y
∂
z
In other coordinate systems in which a point
P
has coordinates
u
,
v
, and
w
, where
we can write
(
)
(
)
(
)
=
uxyz
,,
,
=
vxyz
,,
, and
wwxyz
=
,,
, the gradient operator
u
v
follows the chain rule to become
∂
∂
∂
∇
′ =
∇∇∇
u
+
v
+
w
.
(C.3)
∂
u
∂
v
∂
w
C.2 Cylindrical Coordinates
A point
P
is represented by the quantities
r
,
θ
, and
z
in cylindrical coordinates.
As shown in Figure C.2,
r
is equal to the radial distance between
P
and the
z
axis.
The angle
θ
is called the
azimuthal angle
, or simply the
azimuth
, and is equal to
the counterclockwise angle formed between the
x
axis and the line connecting the
projection of
P
onto the
x

y
plane to the origin. The
z
coordinate has the same
meaning as it does in Cartesian coordinates.
k
P
z
r
j
θ
i
Figure C.2.
Cylindrical coordinates.
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