Geology Reference
In-Depth Information
Fig. 2.2
Cartesian (
x
,
y
,
z
)
and spherical (
r
,™,¥)
coordinates of a point
P
in
the geographic reference
frame
meridian and 90
ı
E. Clearly, a point in the city
of London, on the Eurasian plate, has constant
longitude ¥
D
0 in this reference frame (Fig.
2.2
).
In plate kinematics, the Earth is assumed to
have a spherical shape, so that the Cartesian
coordinates (
x
,
y
,
z
) of a point at distance
r
from
the Earth's centre are related to the geographic
coordinates (™,¥), colatitude and longitude, by
the following equations:
8
<
Another useful geocentric reference frame is
the
geomagnetic coordinate system
(e.g., Camp-
bell
2003
). This frame is built on the basis of the
observation that the present day Earth's magnetic
field can be approximated as the field generated
by a magnetic dipole placed at the Earth's centre,
not fixed direction, but precedes irregularly about
the North Pole according to the so-called
secular
variation
of the core field. It is mathematically
represented by a magnetic moment vector,
m
,
which currently (December 31st 2013) points to
a location placed in the southern hemisphere, at
about (80.24
ı
S, 107.46
ı
E). This location is called
the
geomagnetic South Pole
, and its antipodal
point at (80.24
ı
N, 72.54 W) is known as the
ge-
omagnetic North Pole
. The axis passing through
these two points defines the
z
-axis of the geomag-
netic reference frame. The
x
-axis of this coordi-
nate system is chosen in such a way that the prime
meridian passes through the geographic South
Pole. Finally, the
y
-axis will be also placed in the
geomagnetic dipole equator, 90
ı
from the
x
-axis.
x
D
r sin ™ cos ¥
y
D
r sin ™ sin
¥
z
D
r cos ™
(2.27)
:
Figure
2.2
illustrates the relation between
Cartesian and geographic (spherical) coordinates
of a point. Equations
2.27
can be easily inverted
to get an expression of the spherical coordinates
as a function of the Cartesian components:
8
<
¥
D
arctan .y=x/
™
D
ar
ccos.
z
=r/
(2.28)
r
D
p
x
2
:
C
y
2
C
z
2
