Game Development Reference
In-Depth 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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