Datum Problems - Map Projections: Cartographic Information Systems - page 553

δe +

E sin Φ

−

E 2 sin 2 Φ 1 −

+ A 1 1

δΦ

A 1

E 2 sin 2 Φ ) 2 ( δe ) 2 +

E 2
cos Φ

− 1 sin Φ

− (21.97)

E 2 sin 2 Φ

1 − (1 −

2 A 1 1

1+2 E 2 tan Φ ( δΦ ) 2 +

E 2
cos Φ

3 E 2 sin 2 Φ
(1 − E 2 sin 2 Φ ) 2

+ 1

− −

+ 1

2 ln 1 − E sin Φ

E sin Φ
1

δaδe

− −

E 2 sin 2 Φ

1+ E sin Φ −

2 A 1 E

E 2 sin 2 Φ ) 2 δΦδe + 1 − E 2

δΦδa +

cos Φ 1

−

E 2 sin 2 Φ

(1 − cos Φ 1 −

+O 3 y ,
y :=
:= y 0 + y 1 + y 2 + y 3 + y 4 + y 5 + y 6 + y 7 + y 8 + (21.98)

+O 3 y .

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

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

− E , the variation of the ellipsoidal longitude λ −

A 1 , 4 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 =2 eδe} 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

a 3 ,a 2 to E

, 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 }

{ Λ, Φ, H } .

Box 21.33 (Local coordinates of the universal Mercator projection as a function of global
coordinates and extended datum parameters).

x ( Λ )= A 1 Λ + Λδa + A 1 ( a 11 t x + a 12 t y + a 14 α + a 15 β − γ )+ δaδΛ, (21.99)

y ( Φ )= A 1 ln tan π

E/ 2 + 1

4 + Φ

E sin Φ
1+ E sin Φ

−

2

+ A 1 1 − E 2
cos Φ

1

E 2 sin 2 Φ 1 −