490 21 Datum problems
21-5 Mercator coordinates: from a global to a local datum
Transformation of conformal coordinates of type Mercator from a global datum (WGS 84) to a local
datum (regional, National, European).
The equations which govern the datum transformation in the extended form of parameters of the
Universal Mercator Projection (UMP) are discussed here.
Section 21-51.
In Section 21-51, the basic equations are reviewed: compare with Box 21.33 and Table 21.11.
Section 21-52.
In Section 21-52, a numerical exampls is presented: compare with Tables 21.12–21.19.
21-51 Datum transformation extended by form parameters of the UMP
Let us refer to Definition 21.2 as the universal Mercator projection of the ellipsoid-of-revolution E
2
a 1 ,a 2
in local coordinates, namely ellipsoidal coordinates in a local datum.
Definition 21.2 (Universal Mercator projection, local coordinates).
A conformal transformation of ellipsoidal coordinates of type “surface normal” ellipsoidal longitude λ
and “surface normal” ellipsoidal latitude ϕ into Cartesian coordinates {x, y} with respect to a local
ellipsoid-of-revolution E
2
a 1 ,a 2
is called a universal Mercator projection if (21.94) holds, where a 1 denotes
the semi-major axis, a 2 the semi-minor axis, and e =
1 − a 2
2 /a 2
1 the relative eccentricity of E
2
a 1 ,a 2 .
x = a 1 λ ,
y = a 1 ln
tan
π
4
+
ϕ
2
1 − e sin ϕ
1 + e sin ϕ
e/2
.
(21.94)
End of Definition.
In order to transform Mercator coordinates which are given in a global datum with respect to the
ellipsoid-of-revolution E
2
A 1 ,A 2
into Mercator coordinates which are given in a local datum with respect
to the ellipsoid-of-revolution E
2
a 1 ,a 2 , we take advantage of the Taylor expansion of second order, namely
a 1 = A 1 + δa , e = E + δe ,
λ = Λ + δΛ , ϕ = Φ + δΦ
(21.95)
so that
x(λ, a 1 ) = x(Λ + δΛ, Φ + δΦ, A 1 + δA) = A 1 Λ + Λδa + A 1 δΛ + δAδΛ + O 3x ,
x := x 0 + x 1 + x 2 + x 3 + O 3x
(21.96)
and
21-5 Mercator coordinates: from a global to a local datum
Transformation of conformal coordinates of type Mercator from a global datum (WGS 84) to a local
datum (regional, National, European).
The equations which govern the datum transformation in the extended form of parameters of the
Universal Mercator Projection (UMP) are discussed here.
Section 21-51.
In Section 21-51, the basic equations are reviewed: compare with Box 21.33 and Table 21.11.
Section 21-52.
In Section 21-52, a numerical exampls is presented: compare with Tables 21.12–21.19.
21-51 Datum transformation extended by form parameters of the UMP
Let us refer to Definition 21.2 as the universal Mercator projection of the ellipsoid-of-revolution E
2
a 1 ,a 2
in local coordinates, namely ellipsoidal coordinates in a local datum.
Definition 21.2 (Universal Mercator projection, local coordinates).
A conformal transformation of ellipsoidal coordinates of type “surface normal” ellipsoidal longitude λ
and “surface normal” ellipsoidal latitude ϕ into Cartesian coordinates {x, y} with respect to a local
ellipsoid-of-revolution E
2
a 1 ,a 2
is called a universal Mercator projection if (21.94) holds, where a 1 denotes
the semi-major axis, a 2 the semi-minor axis, and e =
1 − a 2
2 /a 2
1 the relative eccentricity of E
2
a 1 ,a 2 .
x = a 1 λ ,
y = a 1 ln
tan
π
4
+
ϕ
2
1 − e sin ϕ
1 + e sin ϕ
e/2
.
(21.94)
End of Definition.
In order to transform Mercator coordinates which are given in a global datum with respect to the
ellipsoid-of-revolution E
2
A 1 ,A 2
into Mercator coordinates which are given in a local datum with respect
to the ellipsoid-of-revolution E
2
a 1 ,a 2 , we take advantage of the Taylor expansion of second order, namely
a 1 = A 1 + δa , e = E + δe ,
λ = Λ + δΛ , ϕ = Φ + δΦ
(21.95)
so that
x(λ, a 1 ) = x(Λ + δΛ, Φ + δΦ, A 1 + δA) = A 1 Λ + Λδa + A 1 δΛ + δAδΛ + O 3x ,
x := x 0 + x 1 + x 2 + x 3 + O 3x
(21.96)
and