Game Development Reference
In-Depth 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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