10-3 Optimal cylinder projections 279
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
0
0.5
1
1.5
2
2.5
Φ
0
√I
A
Min = 61.72°
Min = 49.31°
55.92°
Conformal mapping
Equal−Area mapping
Equidistant mapping
˛
˛
˛
˛
Fig. 10.7. The Airy optimum of three different mappings: (i) conformal maps, (ii) equiareal maps, and (iii)
distance preserving maps.
10-3 Optimal cylinder projections
Optimal cylinder projections of the sphere of type equidistant on two standard parallels. Conformal
cylindrical mapping, equal area cylindrical mapping, equidistant cylindrical mapping.
Many applications require a map projection the distortions of which do not excess a certain value in
the mean. An example is given by Lemma 10.1.
Lemma 10.1 (Optimal cylinder projections of the sphere of type equidistant on two parallel circles).
If we compare (i) conformal maps, (ii) equiareal maps, and (iii) distance preserving maps in the class
of optimal cylinder projections of the sphere, equidistant on two parallel circles, where the equidistance
on two parallel circles is the unknown parameter, we find according to the Airy optimal criterion that
the distance preserving maps are optimal and the equiareal maps are better than the conformal maps,
at least up to a latitude of Φ = 56
◦ . According to the criterion of Airy–Kavrajski, again the distance
preserving maps are optimal, but the conformal maps and the equiareal maps produce exactly equally
good maps.
End of Lemma.
The two optima of type Airy and Airy–Kavrajski for the mapping of type cylinder projection of the
sphere and equidistant on two parallel circles is illustrated by Fig. 10.7 and Fig. 10.8. Reference Papers
are G. B. Airy (1861), N. Francula (1971), E. W. Grafarend (1995), E. W. Grafarend and A. Niermann
(1984), E. W. Grafarend and R. Syffus (1998c), V. Hojovec and L. Jokl (1981), W. Jordan (1875, 1896),
C. Kaltsikis (1980), V. V. Kavrajski (1958).
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
0
0.5
1
1.5
2
2.5
Φ
0
√I
A
Min = 61.72°
Min = 49.31°
55.92°
Conformal mapping
Equal−Area mapping
Equidistant mapping
˛
˛
˛
˛
Fig. 10.7. The Airy optimum of three different mappings: (i) conformal maps, (ii) equiareal maps, and (iii)
distance preserving maps.
10-3 Optimal cylinder projections
Optimal cylinder projections of the sphere of type equidistant on two standard parallels. Conformal
cylindrical mapping, equal area cylindrical mapping, equidistant cylindrical mapping.
Many applications require a map projection the distortions of which do not excess a certain value in
the mean. An example is given by Lemma 10.1.
Lemma 10.1 (Optimal cylinder projections of the sphere of type equidistant on two parallel circles).
If we compare (i) conformal maps, (ii) equiareal maps, and (iii) distance preserving maps in the class
of optimal cylinder projections of the sphere, equidistant on two parallel circles, where the equidistance
on two parallel circles is the unknown parameter, we find according to the Airy optimal criterion that
the distance preserving maps are optimal and the equiareal maps are better than the conformal maps,
at least up to a latitude of Φ = 56
◦ . According to the criterion of Airy–Kavrajski, again the distance
preserving maps are optimal, but the conformal maps and the equiareal maps produce exactly equally
good maps.
End of Lemma.
The two optima of type Airy and Airy–Kavrajski for the mapping of type cylinder projection of the
sphere and equidistant on two parallel circles is illustrated by Fig. 10.7 and Fig. 10.8. Reference Papers
are G. B. Airy (1861), N. Francula (1971), E. W. Grafarend (1995), E. W. Grafarend and A. Niermann
(1984), E. W. Grafarend and R. Syffus (1998c), V. Hojovec and L. Jokl (1981), W. Jordan (1875, 1896),
C. Kaltsikis (1980), V. V. Kavrajski (1958).