21-5 Mercator coordinates: from a global to a local datum 491
y(ϕ, a 1 , e) =
= y(Φ + δΦ, A 1 + δa, E + δe) =
= A 1 ln
tan
π
4
+
Φ
2
1 − E sin Φ
1 + E sin Φ
E/2
+
+ ln
tan
π
4
+
Φ
2
1 − E sin Φ
1 + E sin Φ
E/2
δa + A 1
1
2
ln
1 − E sin Φ
1 + E sin Φ
−
E sin Φ
1 − E 2 sin
2 Φ
δe+
+A 1
1 − E
2
cos Φ
1
1 − E 2 sin
2 Φ
δΦ − A 1
sin Φ
(1 − E 2 sin
2 Φ) 2
(δe)
2 +
+
1
2
A 1
1 − E
2
cos Φ
1 + 2E
2
− 3E
2 sin
2 Φ
(1 − E 2 sin
2 Φ) 2
tan Φ
(δΦ)
2 +
+
1
2
ln
1 − E sin Φ
1 + E sin Φ
−
E sin Φ
1 − E 2 sin
2 Φ
δaδe−
−2A 1 E
cos Φ
(1 − E 2 sin
2 Φ) 2
δΦδe +
1 − E
2
cos Φ
1
1 − E 2 sin
2 Φ
δΦδa+
+O 3y ,
(21.97)
y :=
:= y 0 + y 1 + y 2 + y 3 + y 4 + y 5 + y 6 + y 7 + y 8 +
+O 3y .
(21.98)
δa, δe and δΛ, δΦ, respectively, as increments, account for the variation of the semi-major axis a 1 −
A 1 , the variation of the relative eccentricity e − E, the variation of the ellipsoidal longitude λ − Λ,
and the variation in the ellipsoidal latitude ϕ − Φ under a geodetic datum transformation, namely,
the conformal group C(3), subject to a variation of the form parameters {a 1 → A 1 , e → E} from
E
2
a 1 ,a 2
to E
2
A 1 ,A 2
. Here, we refer to the curvilinear datum transformation, namely {λ → Λ, ϕ → Φ}.
As soon as we implement the curvilinear datum transformation extended by the ellipsoidal form
parameters {δa, δe
2 = 2eδe} in (21.97), we arrive at the linear representation of local coordinates of
the universal Mercator projection as a function of global coordinates and extended datum parameters
of Box 21.33. Note that the algorithmic version of the datum transformation of UMP coordinates is
given by Table 21.11. Assume that we have measured the ellipsoidal coordinates of a point by means
of {Λ, Φ, H}, for instance, by satellite positioning technology of type GPS, GLONASS, or other. First,
for the synthesis of the design matrix A, we need the global ellipsoidal height. Second, we have to
get information of the variation of the seven datum parameters and the two form parameters, namely
about the basic data which established a local and a global UMP chart. Finally, by means of (21.99)
and (21.100), we are able to compute Easting x(Λ) and Northing y(Φ), namely local UMP coordinates
from global ellipsoidal coordinates {Λ, Φ, H}.