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u i
(up, zenith)
P j
X ij
'
z ij
s ij
u ij
P i
n i
A ij
(north)
e ij
n ij
e i
(east)
Fig. 5.12. Measurement quantities in the local level system
direction, are thus represented in the global system by
sin Φ i cos Λ i
sin Λ i
cos Λ i
0
cos Φ i cos Λ i
cos Φ i sin Λ i
sin Φ i
,
,
,
n i =
sin Φ i sin Λ i
cos Φ i
e i =
u i =
(5-64)
where the vectors n i and e i span the tangent plane at P i (Fig. 5.11). The
third coordinate axis of the local level system, i.e., the vector u i , is orthog-
onal to the tangent plane and has the direction of the plumb line.
Now the components n ij ,e ij ,u ij of the vector x ij in the local level system
are introduced. These coordinates are sometimes denoted as ENU (east,
north, up) coordinates. Considering Fig. 5.12, these components are obtained
by a projection of vector X ij onto the local level axes n i , e i , u i . Analytically,
this is achieved by scalar products. Therefore,
n ij
e ij
u ij
n i ·
X ij
e i · X ij
u i ·
=
x ij =
(5-65)
X ij
is obtained. Assembling the vectors n i , e i , u i
of the local level system as
columns in an orthogonal matrix D i , i.e.,
sin Φ i cos Λ i
sin Λ i
cos Φ i cos Λ i
,
D i =
sin Φ i sin Λ i
cos Λ i
cos Φ i sin Λ i
(5-66)
cos Φ i
0
sin Φ i
relation (5-65) may be written concisely as
x ij = D i X ij .
(5-67)
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