Datum Problems - Map Projections: Cartographic Information Systems - page 520

21-2 Synthesis of a Datum Problem

Datum transformation, datum parameters (translation, rotation, scale). Synthesis matrix,
local heights. Jacobi matrix, chain Jacobi matrix.

Synthesis is understood as the determination of curvilinear global coordinates of points from
curvilinear local coordinates based upon given datum parameters. Since the datum parameters
are close to the identity, Box 21.6 collects the formulae for the computation of ellipsoidal longitude
L and ellipsoidal latitude B sought for in the global reference system from known ellipsoidal
longitude l and ellipsoidal latitude b and given datum parameters t x t y ,t z ,α,β,γ,s,δa,δe 2 }

by a Taylor series expansion. In particular, Box 21.7 highlights the computation of the synthesis
matrix B. Since in the local reference system pseudo-observations of ellipsoidal heights h ( l,b )are
not available, in general, we here study by Box 21.8 the impact of local heights on the synthesis
matrix B, namely by the decomposition B = B 0 + h B 1 .

{

Box 21.6 (Inverse coordinate conformal transformation (datum transformation) extended by
ellipsoid parameters close to the identity).

Inverse transformation { X, Y, Z }→{ L, B, H } :

L = arctan( Y/X )+

2 sgn Y sgn X +1 π ∈{ R | 0 ≤ L< 2 τ},

1
2 sgn Y −

1

−

B = arctan Z

=

E 2 sin 2 B ) − 1 / 2 + H ( L, B ) A 1 (1 −

√ X 2 + Y 2 (21.31)

E 2 sin 2 B ) − 1 / 2 + H ( L, B )

A 1 (1 − E 2 )(1 −

= arctan Z

N ( B )+ H ( L, B )

√ X 2 + Y 2

(1 − E 2 ) N ( B )+ H ( L, B )

.
Inverse curvilinear coordinate conformal transformation close to
the identity (datum transformation) .
1st variant :

∈{ R |− π/ 2 <B< + π/ 2 }

L = arctan( y − t y − zα + xγ − ys ) / ( x − t z + z/ 3 − yγ − xs ) ,

(21.32)

B = B ( x, y, z, t x ,t y ,t z ,α,β,γ,s ) .
2nd variant:

tan L =

= [ a 1 / 1 − e 2 sin 2 b + h ( l,b )] cos b sin l − t y − [ a 1 (1 − e 2 ) / 1 − e 2 sin 2 b
+ h ( l,b )] sin bα