Gravity field of the earth - Physical Geodesy - page 51

2.4 Natural coordinates

The system of level surfaces and plumb lines may be used as a three-
dimensional curvilinear coordinate system that is well suited to certain pur-
poses; these coordinates can be measured directly, as opposed to local rectan-
gular coordinates x, y, z . Note, however, that global rectangular coordinates
may be measured directly using satellites, see Sect. 5.3.
The direction of the earth's axis of rotation and the position of the equa-
torial plane (normal to the axis) are well defined astronomically. The astro-
nomical latitude Φofapoint P is the angle between the vertical (direction
of the plumb line) at P and the equatorial plane, see Fig. 2.7. From this
figure, we also see that line PN is parallel to the rotation axis, plane GP F
normal to it, that is, parallel to the equatorial plane; n is the unit vector
along the plumb line; plane NPF is the meridian plane of P , and plane
NPG is parallel to the meridian plane of Greenwich.
Consider now a straight line through P parallel to the earth's axis of
rotation. This parallel and the vertical at P together define the meridian
plane of P . The angle between this meridian plane and the meridian plane
of Greenwich (or some other fixed plane) is the astronomical longitude Λof
P . (Exercise: define Φ and Λ without using the unit sphere. The solution
may be found in Sect. 5.9).

unit sphere

N

vertical

n

P

G

F

uato r

earth

Fig. 2.7. Definition of the astronomical coordinates Φ and Λ of P by
means of a unit sphere with center at P