21-4 Gauss–Krueger/UTM coordinates: from a local to a global datum 475
Box 21.14 (Polynomial representation of conformal coordinates of type Gauss–Krueger or UTM after a curvilinear datum transformation).
X(l − l 0 , b − b 0 ) =
= ρ
h
U 00 + U 10 (l − l 0 ) + U 01 (b − b 0 ) + U 20 (l − l 0 )
2 + U 11 (l − l 0 )(b − b 0 ) + U 02 (b − b 0 )
2 +
+U 30 (l − l 0 )
3 + U 21 (l − l 0 )
2 (b − b 0 ) + U 12 (l − l 0 )(b − b 0 )
2 + U 03 (b − b 0 )
3
i
+
+O(4) ,
Y (l − l 0 , b − b 0 ) =
= ρ
h
V 00 + V 10 (l − l 0 ) + V 01 (b − b 0 ) + V 20 (l − l 0 )
2 + V 11 (l − l 0 )(b − b 0 ) + V 02 (b − b 0 )
2 +
+V 30 (l − l 0 )
3 + V 21 (l − l 0 )
2 (b − b 0 ) + V 12 (l − l 0 )(b − b 0 )
2 + V 03 (b − b 0 )
3
i
+
+O(4) .
(21.70)
Box 21.15 (Polynomial coefficients of a conformal series of type Gauss–Krueger or UTM after a curvilinear
datum transformation, Easting X(l − l 0 , b − b 0 ), first version).
U 00 = X 10 δL + X 11 δLδB ,
U 10 = X 10 + X 11 δB ,
U 01 = X 11 δL ,
U 20 = 3X 30 δL ,
U 11 = X 11 + 2X 12 δB , U 02 = X 12 δL ,
U 30 = X 30 + X 31 δB ,
U 21 = 3X 31 δL ,
U 12 = X 12 + 3X 13 δB , U 03 = X 13 δL .
(21.71)
Box 21.16 (Polynomial coefficients of a conformal series of type Gauss–Krueger or UTM after a curvilinear
datum transformation, Northing Y (l − l 0 , b − b 0 ), first version).
V 00 = Y 00 + Y 01 δL + Y 20 δL
2 + Y 02 δB
2 ,
V 10 = 2Y 20 δL ,
V 01 = Y 01 + 2Y 02 δB ,
V 20 = Y 20 + Y 21 δB ,
V 11 = 2Y 21 δL ,
V 02 = Y 02 + 3Y 03 δB ,
V 30 = 4Y 40 δL ,
V 21 = Y 21 + 2Y 22 δB , V 12 = 2Y 22 δL ,
V 03 = Y 03 + 4Y 04 δB .
(21.72)
Box 21.17 (Transformation of the coefficients of a conformal series under a curvilinear datum transformation,
including the change of form parameter by δE
2 , transformation close to the identity).
X 10 =
A 1
a 1
`
x 10 + x 10,e 2 δE
2 ´
, X 12 =
A 1
a 1
`
x 12 + x 12,e 2 δE
2 ´
,
X 11 =
A 1
a 1
`
x 11 + x 11,e 2 δE
2 ´
, X 30 =
A 1
a 1
`
x 30 + x 30,e 2 δE
2 ´
,
(21.73)
Y 00 =
A 1
a 1
„
y 00 + y 00,e 2 δE
2 + y 00,e 2 e 2
(δE
2 )
2
2
«
,
Y 01 =
A 1
a 1
`
y 01 + y 01,e 2 δE
2 ´
, Y 02 =
A 1
a 1
`
y 02 + y 02,e 2 δE
2 ´
,
Y 20 =
A 1
a 1
`
y 20 + y 20,e 2 δE
2 ´
, Y 21 =
A 1
a 1
`
y 21 + y 21,e 2 δE
2 ´
,
Y 03 =
A 1
a 1
`
y 03 + y 03,e 2 δE
2 ´
.
(21.74)
Box 21.14 (Polynomial representation of conformal coordinates of type Gauss–Krueger or UTM after a curvilinear datum transformation).
X(l − l 0 , b − b 0 ) =
= ρ
h
U 00 + U 10 (l − l 0 ) + U 01 (b − b 0 ) + U 20 (l − l 0 )
2 + U 11 (l − l 0 )(b − b 0 ) + U 02 (b − b 0 )
2 +
+U 30 (l − l 0 )
3 + U 21 (l − l 0 )
2 (b − b 0 ) + U 12 (l − l 0 )(b − b 0 )
2 + U 03 (b − b 0 )
3
i
+
+O(4) ,
Y (l − l 0 , b − b 0 ) =
= ρ
h
V 00 + V 10 (l − l 0 ) + V 01 (b − b 0 ) + V 20 (l − l 0 )
2 + V 11 (l − l 0 )(b − b 0 ) + V 02 (b − b 0 )
2 +
+V 30 (l − l 0 )
3 + V 21 (l − l 0 )
2 (b − b 0 ) + V 12 (l − l 0 )(b − b 0 )
2 + V 03 (b − b 0 )
3
i
+
+O(4) .
(21.70)
Box 21.15 (Polynomial coefficients of a conformal series of type Gauss–Krueger or UTM after a curvilinear
datum transformation, Easting X(l − l 0 , b − b 0 ), first version).
U 00 = X 10 δL + X 11 δLδB ,
U 10 = X 10 + X 11 δB ,
U 01 = X 11 δL ,
U 20 = 3X 30 δL ,
U 11 = X 11 + 2X 12 δB , U 02 = X 12 δL ,
U 30 = X 30 + X 31 δB ,
U 21 = 3X 31 δL ,
U 12 = X 12 + 3X 13 δB , U 03 = X 13 δL .
(21.71)
Box 21.16 (Polynomial coefficients of a conformal series of type Gauss–Krueger or UTM after a curvilinear
datum transformation, Northing Y (l − l 0 , b − b 0 ), first version).
V 00 = Y 00 + Y 01 δL + Y 20 δL
2 + Y 02 δB
2 ,
V 10 = 2Y 20 δL ,
V 01 = Y 01 + 2Y 02 δB ,
V 20 = Y 20 + Y 21 δB ,
V 11 = 2Y 21 δL ,
V 02 = Y 02 + 3Y 03 δB ,
V 30 = 4Y 40 δL ,
V 21 = Y 21 + 2Y 22 δB , V 12 = 2Y 22 δL ,
V 03 = Y 03 + 4Y 04 δB .
(21.72)
Box 21.17 (Transformation of the coefficients of a conformal series under a curvilinear datum transformation,
including the change of form parameter by δE
2 , transformation close to the identity).
X 10 =
A 1
a 1
`
x 10 + x 10,e 2 δE
2 ´
, X 12 =
A 1
a 1
`
x 12 + x 12,e 2 δE
2 ´
,
X 11 =
A 1
a 1
`
x 11 + x 11,e 2 δE
2 ´
, X 30 =
A 1
a 1
`
x 30 + x 30,e 2 δE
2 ´
,
(21.73)
Y 00 =
A 1
a 1
„
y 00 + y 00,e 2 δE
2 + y 00,e 2 e 2
(δE
2 )
2
2
«
,
Y 01 =
A 1
a 1
`
y 01 + y 01,e 2 δE
2 ´
, Y 02 =
A 1
a 1
`
y 02 + y 02,e 2 δE
2 ´
,
Y 20 =
A 1
a 1
`
y 20 + y 20,e 2 δE
2 ´
, Y 21 =
A 1
a 1
`
y 21 + y 21,e 2 δE
2 ´
,
Y 03 =
A 1
a 1
`
y 03 + y 03,e 2 δE
2 ´
.
(21.74)