348 15 “Ellipsoid-of-revolution to cylinder”: transverse aspect
15-611 The first step: polynomial representation of conformal coordinates in the first strip and
bivariate series inversion
The standard polynomial representation of conformal coordinates of type Gauss–Krueger or UTM
in the L 01 -strip is given by (15.109) and (15.110) subject to the longitude/latitude differences
l 1 := L − L 01 and b 1 := B − B 01 with respect to the longitude L 01 of the reference meridian and
the latitude B 01 of the reference point {L 01 , B 01 } of series expansion.
Easting:
x 1 = ρ
x 10 l 1 + x 11 l 1 b 1 + x 30 l
3
1 + x 12 l 1 b
2
1 + O 4x
.
(15.109)
Northing:
y 1 = ρ
y 0 + y 01 b 1 + y 20 l
2
1 + y 02 b
2
1 + y 03 b
3
1 + O 4y
.
(15.110)
y 0 denotes the length of the meridian arc from zero ellipsoidal latitude to the ellipsoidal latitude B 01
of the reference point {L 01 , B 01 }. The dilatation factor ρ amounts to one for a classical Gauss–Krueger
conformal mapping. Optimal alternative values for the dilatation factor depending on the width of the
strip, namely for UTM, are given in Box 15.9. The coefficients {x ij , y ij } of the conformal polynomial
of type (15.109) and (15.110) of order five are derived in E. Grafarend (1995, p. 457–459), for instance,
and listed in Boxes 15.4 and 15.5. The length y 0 of the meridian arc from the equator to the reference
point is computed from (15.114) in Box 15.10.
Box 15.9 (Optimal dilatation factor for a Universal Transverse Mercator mapping of an ellipsoid-of-revolution
E
2
A 1 ,A 2 according to E. Grafarend (1995 p. 459–461), l E := L − L 0 eastern longitude difference, B S and B N
southern latitude and northern latitude).
Strip width [−l E , l E ] × [B S , B N ]:
Optimal dilatation factor:
[−3.5
◦ , +3.5
◦ ] × [80
◦ , 84
◦ ] ,
0.999 578 ,
[−2
◦ , +2
◦ ] × [80
◦ , 80
◦ ] .
0.999 864 .
(15.111)
As outlined by E. Grafarend, T. Krarup and R. Syffus (1996, p. 279–284), the inversion of the bivariate homogeneous conformal polynomial (15.109) and (15.110) leads us to the bivariate homogenous
polynomial (15.112) and (15.113) with coefficients {l ij , b ij } summarized in Box 15.11.
l 1 = L − L 01 =
= l 10
x 1
ρ
+ l 11
x 1
ρ
y 1
ρ
− y 0
+ l 30
x 1
ρ
3
+
+l 12
x 1
ρ
y 1
ρ
− y 0
2
+ O 4l ,
(15.112)
b 1 = B − B 01 =
= b 01
y 1
ρ
− y 0
+ b 20
x 1
ρ
2
+ b 02
y 1
ρ
− y 0
2
+
+b 21
x 1
ρ
2
y 1
ρ
− y 0
+ b 03
y 1
ρ
− y 0
3
+ O 4b .
(15.113)
15-611 The first step: polynomial representation of conformal coordinates in the first strip and
bivariate series inversion
The standard polynomial representation of conformal coordinates of type Gauss–Krueger or UTM
in the L 01 -strip is given by (15.109) and (15.110) subject to the longitude/latitude differences
l 1 := L − L 01 and b 1 := B − B 01 with respect to the longitude L 01 of the reference meridian and
the latitude B 01 of the reference point {L 01 , B 01 } of series expansion.
Easting:
x 1 = ρ
x 10 l 1 + x 11 l 1 b 1 + x 30 l
3
1 + x 12 l 1 b
2
1 + O 4x
.
(15.109)
Northing:
y 1 = ρ
y 0 + y 01 b 1 + y 20 l
2
1 + y 02 b
2
1 + y 03 b
3
1 + O 4y
.
(15.110)
y 0 denotes the length of the meridian arc from zero ellipsoidal latitude to the ellipsoidal latitude B 01
of the reference point {L 01 , B 01 }. The dilatation factor ρ amounts to one for a classical Gauss–Krueger
conformal mapping. Optimal alternative values for the dilatation factor depending on the width of the
strip, namely for UTM, are given in Box 15.9. The coefficients {x ij , y ij } of the conformal polynomial
of type (15.109) and (15.110) of order five are derived in E. Grafarend (1995, p. 457–459), for instance,
and listed in Boxes 15.4 and 15.5. The length y 0 of the meridian arc from the equator to the reference
point is computed from (15.114) in Box 15.10.
Box 15.9 (Optimal dilatation factor for a Universal Transverse Mercator mapping of an ellipsoid-of-revolution
E
2
A 1 ,A 2 according to E. Grafarend (1995 p. 459–461), l E := L − L 0 eastern longitude difference, B S and B N
southern latitude and northern latitude).
Strip width [−l E , l E ] × [B S , B N ]:
Optimal dilatation factor:
[−3.5
◦ , +3.5
◦ ] × [80
◦ , 84
◦ ] ,
0.999 578 ,
[−2
◦ , +2
◦ ] × [80
◦ , 80
◦ ] .
0.999 864 .
(15.111)
As outlined by E. Grafarend, T. Krarup and R. Syffus (1996, p. 279–284), the inversion of the bivariate homogeneous conformal polynomial (15.109) and (15.110) leads us to the bivariate homogenous
polynomial (15.112) and (15.113) with coefficients {l ij , b ij } summarized in Box 15.11.
l 1 = L − L 01 =
= l 10
x 1
ρ
+ l 11
x 1
ρ
y 1
ρ
− y 0
+ l 30
x 1
ρ
3
+
+l 12
x 1
ρ
y 1
ρ
− y 0
2
+ O 4l ,
(15.112)
b 1 = B − B 01 =
= b 01
y 1
ρ
− y 0
+ b 20
x 1
ρ
2
+ b 02
y 1
ρ
− y 0
2
+
+b 21
x 1
ρ
2
y 1
ρ
− y 0
+ b 03
y 1
ρ
− y 0
3
+ O 4b .
(15.113)