15 “Ellipsoid-of-revolution to cylinder”: transverse aspect
315
Section 15-3.
Section 15-3, in contrast, outlines the constraints to the general solution of the Korn–Lichtenstein
equations subject to the integrability conditions of type Laplace–Beltrami, which lead directly to the
conformal coordinates of type Gauss–Krueger or UTM. Such a solution is generated by the equidistant
mapping of the meridian of reference L 0 , for UTM up to a dilatation factor, as the proper constraint
(x(0, b) = 0 and y(0, b) given). The highlight is the theorem which gives the solution of the partial
differential equations for the conformal mapping in terms of a conformal set of bivariate polynomials.
Throughout, we use a right-handed coordinate system, namely x “Easting” and y “Northing”. Box 15.4
and Box 15.5 contain the non-vanishing polynomial coefficients in a closed form.
Section 15-4.
Section 15-4 introduces by four corollaries the left Cauchy–Green tensor and the dilatation factor for
both the UTM reference frame as well as the Gauss–Krueger reference frame with the values (15.2)
and (15.3) based upon the geometry of the “Geodetic Reference System 1980” (H. ˙
Moritz 1984). Such
a result was achieved by (i) minimizing the total distance distortion or (ii) minimizing the total areal
distortion with the identical result.
UTM:
[−l E , +l E ] × [B S , B N ] = [−3.5
◦ , +3.5
◦ ] × [80
◦ S, 84
◦ N] ,
ρ = 0.999 578
(scale reduction factor 1 : 2 370) ,
(15.2)
Gauss–Krueger:
[−l E , +l E ] × [B S , B N ] = [−2
◦ , +2
◦ ] × [80
◦ S, 80
◦ N] ,
ρ = 0.999 864
(scale reduction factor 1 : 7 353) .
(15.3)
(The symbols S, N, E, and W as indices denote South, North, East, and West.)
Section 15-5.
Examples are the subject of Section 15-5. In particular, compare with Figs. 15.6–15.16 dealing with
the transverse Mercator projection.
Section 15-6.
Strip transformations of conformal coordinates of type Gauss–Krueger as well as of type UTM are
finally the subject of Section 15-6.
Appendix
In Appendix D, we outline the theory of the Cauchy–Green deformation tensor and its related general
eigenvalue–eigenvector problem, in particular, its conformal structure, which leads us to three forms
of the related Korn–Lichtenstein equations.
315
Section 15-3.
Section 15-3, in contrast, outlines the constraints to the general solution of the Korn–Lichtenstein
equations subject to the integrability conditions of type Laplace–Beltrami, which lead directly to the
conformal coordinates of type Gauss–Krueger or UTM. Such a solution is generated by the equidistant
mapping of the meridian of reference L 0 , for UTM up to a dilatation factor, as the proper constraint
(x(0, b) = 0 and y(0, b) given). The highlight is the theorem which gives the solution of the partial
differential equations for the conformal mapping in terms of a conformal set of bivariate polynomials.
Throughout, we use a right-handed coordinate system, namely x “Easting” and y “Northing”. Box 15.4
and Box 15.5 contain the non-vanishing polynomial coefficients in a closed form.
Section 15-4.
Section 15-4 introduces by four corollaries the left Cauchy–Green tensor and the dilatation factor for
both the UTM reference frame as well as the Gauss–Krueger reference frame with the values (15.2)
and (15.3) based upon the geometry of the “Geodetic Reference System 1980” (H. ˙
Moritz 1984). Such
a result was achieved by (i) minimizing the total distance distortion or (ii) minimizing the total areal
distortion with the identical result.
UTM:
[−l E , +l E ] × [B S , B N ] = [−3.5
◦ , +3.5
◦ ] × [80
◦ S, 84
◦ N] ,
ρ = 0.999 578
(scale reduction factor 1 : 2 370) ,
(15.2)
Gauss–Krueger:
[−l E , +l E ] × [B S , B N ] = [−2
◦ , +2
◦ ] × [80
◦ S, 80
◦ N] ,
ρ = 0.999 864
(scale reduction factor 1 : 7 353) .
(15.3)
(The symbols S, N, E, and W as indices denote South, North, East, and West.)
Section 15-5.
Examples are the subject of Section 15-5. In particular, compare with Figs. 15.6–15.16 dealing with
the transverse Mercator projection.
Section 15-6.
Strip transformations of conformal coordinates of type Gauss–Krueger as well as of type UTM are
finally the subject of Section 15-6.
Appendix
In Appendix D, we outline the theory of the Cauchy–Green deformation tensor and its related general
eigenvalue–eigenvector problem, in particular, its conformal structure, which leads us to three forms
of the related Korn–Lichtenstein equations.