Game Development Reference
In-Depth Information
Coordinate System
Metric Tensor
100
010
001
[
]
g
=
Cartesian coordinates
ij
100
0 0
001
[
]
2
g
=
r
Cylindrical coordinates
ij
1 0 0
0sin 0
0
[
]
g
=
r
2
2
φ
Spherical coordinates
ij
0
r
2
Table C.1. Metric tensors.
This establishes a metric in the coordinate system S and reveals the source of the
metric tensor's name. If the vector v represents the coordinate difference between
two points, then the metric tensor is used in Equation (C.27) to obtain a (not gen-
erally Euclidean) measure of distance between the two points.
To calculate the Euclidean distance between two points, we integrate differ-
ential distances along a straight-line path. Straight lines are not generally given
by linear functions of the coordinates, so we consider an arbitrary parametric
path
()
t
in the coordinate system S . The length L of the path over the interval in
u
]
which
t
[
ab
,
is given by
b
b
12
 
d
u
d
u
d
u
L
=
dt
=
dt
dt
dt
dt
a
a
b
12
33
du
du

i
j
=
g
dt
.
(C.28)
ij
dt
dt
i
==
11
j
a
The quantity
33

ds
2
=
g
du du
(C.29)
ij
i
j
i
==
11
j