334 15 “Ellipsoid-of-revolution to cylinder”: transverse aspect
For this chapter and the other chapters, please consult the following publications. G. B. Airy
(1861), M. Amalvict and E. Livieratos (1988), E. Beltrami (1869), W. Blaschke and K. Leichtweiß
(1973), K. Bretterbauer (1980), M. do Carmo, M. Dajczer, and F. Mercuri (1985), A. Cauchy (1823,
1828), A. R. Clarke and F. R. Helmert (1911), J. H. Cole (1943), D. Cox, J. Little, and D. O’Shea
(1996), A. Dermanis and E. Livieratos (1983, 1993), A. Dermanis, E. Livieratos, and S. Pertsinidou
(1984), J. Engels and E. Grafarend (1995), L. Euler (1755, 1770), A. Finzi (1922), C. F. Gauss
(1813, 1816–1827, 1822, 1844), H. Glasmacher, K. Krack (1984), H. Goenner, E. Grafarend, and
R. J. You (1994), E. Grafarend (1995), E. Grafarend and R. Syffus (1995, 1998c), G. Green (1841),
E. R. Hedrick and L. Ingold (1925a,b), S. Heitz (1988), M. Hotine (1946, 1947), C. G. J. Jacobi (1839),
C. Kaltsikis (1980), V. V. Kavrajski (1958), W. Klingenberg (1982), R. K¨ onig and K. H. Weise (1951),
A. Korn (1914), L. Krueger (1903, 1912, 1914, 1922), R. S. Kulkarni (1969, 1972), R. S. Kulkarni
and U. Pinkall (eds. 1988), Laborde (1928), J. Lafontaine (1988a,b), J. L. Lagrange (1781), L. P. Lee
(1944, 1976), L. Lichtenstein (1911, 1916), R. Lilienthal (1902–1927), J. Liouville (1850), E. Livieratos
(1987), C. F. van Loan (1976), D. H. Maling (1960, 1973), H. Maurer (1935), A. I. Markuschewitsch
(1955), O. M. Miller (1941), C. W. Misner (1978), S. K. Mitra and C. R. Rao (1968), B. de Moor
and H. Zha (1991), B. Mueller (1991), G. Ricci (1918), P. Richardus and R. K. Adler (1972a,b),
C. F. B. Riemann (1851), M. Rosenmund (1903), E. Schering (1857), H. Schmehl (1927), J. A. Schouten
(1921), J. P. Snyder (1979a–c, 1982), K. Spallek (1980), E. M. Stein and G. Weiss (1968), T. C. T. Ting
(1985), N. A. Tissot (1881), F. Uhlig (1976, 1979), H. Weber (1867), T. Wray (1974), K. Yano (1970),
A. I. Yanushaushas (1982), M. Zadro and A. Carminelli (1966), H. Zha (1991) and J. Zund (1987).
15-5 Examples (Gauss–Krueger/UTM coordinates)
Various interesting Examples. Mapping of the transverse Mercator projection. Gauss–Krueger/UTM coordinates. Strip system, meridian strip system of Germany.
There has been the result that the regular transverse Mercator projection of the sphere is simple and
its mathematical version does not cause any problem. The picture changes if we move to the transverse
Mercator projection of the ellipsoid-of-revolution.
Important!
It relates to the elliptical transverse cylinder. It is conformal. Its central meridian and each
meridian 90
◦ apart from it are straight lines. Its equator is a straight line, other meridians and
parallels are complex curves. Scale is true along the central meridian or along two straight
lines in the map equidistant from and parallel to the central meridian, constant along any
straight line on the map parallel to the central meridian. Scale becomes infinite 90
◦ from
the reference meridian. It is used extensively for quadrangle maps at scales from 1 : 25000
to 1 : 250000.
We recall the representation of Transverse Mercator coordinates for the ellipsoid-of-revolution in the
following form.
x(l, b) =
= x 10 l + x 11 lb + x 30 l
3 + x 12 lb
2 + x 31 l
3 b + x 13 lb
3 + x 50 l
5 + x 32 l
3 b
2 + x 14 lb
4 +
+O(6)
(Easting) ,
(15.107)
y(l, b) =
= y 01 b + y 20 l
2 + y 02 b
2 + y 21 l
2 b + y 03 b
3 + y 40 l
4 + y 22 l
2 b
2 + y 04 b
4 + y 41 l
4 b + y 23 l
2 b
3 +
+O(6)
(Northing) .
(15.108)
For this chapter and the other chapters, please consult the following publications. G. B. Airy
(1861), M. Amalvict and E. Livieratos (1988), E. Beltrami (1869), W. Blaschke and K. Leichtweiß
(1973), K. Bretterbauer (1980), M. do Carmo, M. Dajczer, and F. Mercuri (1985), A. Cauchy (1823,
1828), A. R. Clarke and F. R. Helmert (1911), J. H. Cole (1943), D. Cox, J. Little, and D. O’Shea
(1996), A. Dermanis and E. Livieratos (1983, 1993), A. Dermanis, E. Livieratos, and S. Pertsinidou
(1984), J. Engels and E. Grafarend (1995), L. Euler (1755, 1770), A. Finzi (1922), C. F. Gauss
(1813, 1816–1827, 1822, 1844), H. Glasmacher, K. Krack (1984), H. Goenner, E. Grafarend, and
R. J. You (1994), E. Grafarend (1995), E. Grafarend and R. Syffus (1995, 1998c), G. Green (1841),
E. R. Hedrick and L. Ingold (1925a,b), S. Heitz (1988), M. Hotine (1946, 1947), C. G. J. Jacobi (1839),
C. Kaltsikis (1980), V. V. Kavrajski (1958), W. Klingenberg (1982), R. K¨ onig and K. H. Weise (1951),
A. Korn (1914), L. Krueger (1903, 1912, 1914, 1922), R. S. Kulkarni (1969, 1972), R. S. Kulkarni
and U. Pinkall (eds. 1988), Laborde (1928), J. Lafontaine (1988a,b), J. L. Lagrange (1781), L. P. Lee
(1944, 1976), L. Lichtenstein (1911, 1916), R. Lilienthal (1902–1927), J. Liouville (1850), E. Livieratos
(1987), C. F. van Loan (1976), D. H. Maling (1960, 1973), H. Maurer (1935), A. I. Markuschewitsch
(1955), O. M. Miller (1941), C. W. Misner (1978), S. K. Mitra and C. R. Rao (1968), B. de Moor
and H. Zha (1991), B. Mueller (1991), G. Ricci (1918), P. Richardus and R. K. Adler (1972a,b),
C. F. B. Riemann (1851), M. Rosenmund (1903), E. Schering (1857), H. Schmehl (1927), J. A. Schouten
(1921), J. P. Snyder (1979a–c, 1982), K. Spallek (1980), E. M. Stein and G. Weiss (1968), T. C. T. Ting
(1985), N. A. Tissot (1881), F. Uhlig (1976, 1979), H. Weber (1867), T. Wray (1974), K. Yano (1970),
A. I. Yanushaushas (1982), M. Zadro and A. Carminelli (1966), H. Zha (1991) and J. Zund (1987).
15-5 Examples (Gauss–Krueger/UTM coordinates)
Various interesting Examples. Mapping of the transverse Mercator projection. Gauss–Krueger/UTM coordinates. Strip system, meridian strip system of Germany.
There has been the result that the regular transverse Mercator projection of the sphere is simple and
its mathematical version does not cause any problem. The picture changes if we move to the transverse
Mercator projection of the ellipsoid-of-revolution.
Important!
It relates to the elliptical transverse cylinder. It is conformal. Its central meridian and each
meridian 90
◦ apart from it are straight lines. Its equator is a straight line, other meridians and
parallels are complex curves. Scale is true along the central meridian or along two straight
lines in the map equidistant from and parallel to the central meridian, constant along any
straight line on the map parallel to the central meridian. Scale becomes infinite 90
◦ from
the reference meridian. It is used extensively for quadrangle maps at scales from 1 : 25000
to 1 : 250000.
We recall the representation of Transverse Mercator coordinates for the ellipsoid-of-revolution in the
following form.
x(l, b) =
= x 10 l + x 11 lb + x 30 l
3 + x 12 lb
2 + x 31 l
3 b + x 13 lb
3 + x 50 l
5 + x 32 l
3 b
2 + x 14 lb
4 +
+O(6)
(Easting) ,
(15.107)
y(l, b) =
= y 01 b + y 20 l
2 + y 02 b
2 + y 21 l
2 b + y 03 b
3 + y 40 l
4 + y 22 l
2 b
2 + y 04 b
4 + y 41 l
4 b + y 23 l
2 b
3 +
+O(6)
(Northing) .
(15.108)