634 I Mercator projection and polycylindric projection
zone 1
geographic region of Indonesia
zone 0
zone -1
zone -2
+8
◦
+4
◦
0
◦
−4
◦
−8
◦
−12
◦
Λ = 95
◦
Λ = 120
◦
Λ = 145
◦
Fig. I.3. UPC, geographic region of Indonesia, strip [95
◦ < Λ < 145
◦ ] × [−12
◦ < Φ < +8
◦ ] and strip width
Φ N − Φ S = 2∆Φ = 6
◦ , the zones 0, ±1, −2.
Finally, we present as an example the Airy optimal UPC for a strip system which extends to −12
◦
of southerly and +8
◦ of northerly latitude. Again this example can be considered as analogous to one
given by E. Grafarend (1995) for the optimal transverse Mercator projection.
Example I.3 ([95
◦ < Λ < 145
◦ ] × [−12
◦ < Φ < +8
◦ ]).
For the Airy optimal UPC, we have chosen a strip width of 6
◦ between Φ S = −12
◦ and Φ N = 8
◦ of
southern and northern latitude, in particular, to match the geographic region of Indonesia. Once we
refer to WGS 84, the strip system as well as the dilatation factor ˆ
ρ 0 per strip is illustrated by Fig. I.3,
namely for the zones 0, ±1, −2.
End of Example.
Furthermore, we computed by means of (I.34) the left principal stretches Λ 1 = Λ 2 = Λ S (Φ 0 , Φ; ˆ
ρ 0 ) of
each strip and plotted them in Figs. I.4–I.6. Moreover, Tables I.4 and I.5 give the latitude ¯
Φ of each
strip according to the strip width ˆ
ρ 0 and Φ 0 along which the mapping is equidistant. ¯
Φ is determined by
solving (I.34) for given strip width, ˆ
ρ 0 and Φ 0 under the condition that Λ 1 = Λ 2 = 1 holds. Obviously,
the variation of the left principal stretch Λ 1 = Λ 2 = Λ S (Φ 0 , Φ; ˆ
ρ 0 ) is small within the chosen strip.
Alternatively, we may say that the radius of the left Tissot circle Λ 1 = Λ 2 = Λ S (Φ 0 , Φ; ˆ
ρ 0 ) varies only
for a small amount, a favourable result of the Airy optimal UPC.
Important!
For regions with a East–West extension around the equator, the universal Mercator projection is Airy optimal. In contrast, the universal transverse Mercator projection is Airy
optimal if the region extends North–South. The oblique Mercator projection is Airy optimal
for an oblique extension of a region. The analogue statement holds for the Universal Polycylindric Projection (UPS), the Universal Transverse Polycylindric Projection (UTPC), and
the Oblique Polycylindric Projection (OPC).
zone 1
geographic region of Indonesia
zone 0
zone -1
zone -2
+8
◦
+4
◦
0
◦
−4
◦
−8
◦
−12
◦
Λ = 95
◦
Λ = 120
◦
Λ = 145
◦
Fig. I.3. UPC, geographic region of Indonesia, strip [95
◦ < Λ < 145
◦ ] × [−12
◦ < Φ < +8
◦ ] and strip width
Φ N − Φ S = 2∆Φ = 6
◦ , the zones 0, ±1, −2.
Finally, we present as an example the Airy optimal UPC for a strip system which extends to −12
◦
of southerly and +8
◦ of northerly latitude. Again this example can be considered as analogous to one
given by E. Grafarend (1995) for the optimal transverse Mercator projection.
Example I.3 ([95
◦ < Λ < 145
◦ ] × [−12
◦ < Φ < +8
◦ ]).
For the Airy optimal UPC, we have chosen a strip width of 6
◦ between Φ S = −12
◦ and Φ N = 8
◦ of
southern and northern latitude, in particular, to match the geographic region of Indonesia. Once we
refer to WGS 84, the strip system as well as the dilatation factor ˆ
ρ 0 per strip is illustrated by Fig. I.3,
namely for the zones 0, ±1, −2.
End of Example.
Furthermore, we computed by means of (I.34) the left principal stretches Λ 1 = Λ 2 = Λ S (Φ 0 , Φ; ˆ
ρ 0 ) of
each strip and plotted them in Figs. I.4–I.6. Moreover, Tables I.4 and I.5 give the latitude ¯
Φ of each
strip according to the strip width ˆ
ρ 0 and Φ 0 along which the mapping is equidistant. ¯
Φ is determined by
solving (I.34) for given strip width, ˆ
ρ 0 and Φ 0 under the condition that Λ 1 = Λ 2 = 1 holds. Obviously,
the variation of the left principal stretch Λ 1 = Λ 2 = Λ S (Φ 0 , Φ; ˆ
ρ 0 ) is small within the chosen strip.
Alternatively, we may say that the radius of the left Tissot circle Λ 1 = Λ 2 = Λ S (Φ 0 , Φ; ˆ
ρ 0 ) varies only
for a small amount, a favourable result of the Airy optimal UPC.
Important!
For regions with a East–West extension around the equator, the universal Mercator projection is Airy optimal. In contrast, the universal transverse Mercator projection is Airy
optimal if the region extends North–South. The oblique Mercator projection is Airy optimal
for an oblique extension of a region. The analogue statement holds for the Universal Polycylindric Projection (UPS), the Universal Transverse Polycylindric Projection (UTPC), and
the Oblique Polycylindric Projection (OPC).