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Lemma I.12 (Minimal Airy distortion energy).
The Airy distortion energy ( I.37 ) is minimal if the dilatation factor amounts to ρ 0 = c 01 /c 02 and
ρ 0 ( Φ S N )=
= ρ 0 ( Φ S N ) 1
E 2 sin 2 Φ 0
cos Φ 0
,
ρ 0 ( Φ S =
Φ N ) =
(I.39)
= ρ 0 ( Φ S = −Φ N ) 1 E 2 sin 2 Φ 0
cos Φ 0
,
where c 01 and c 02 follow from ( I.38 )and( I.10 ), ρ 0 ( Φ S = Φ N )from( I.19 ) (see the first equation),
and ρ 0 ( Φ S = −Φ N )from( I.19 ) (see the second equation), respectively.
End of Lemma.
The proof of Lemma I.12 completely follows along the lines of the proof of Lemma I.5 and is
therefore not repeated here. In addition, we note zero total areal distortion over a half-symmetric
strip [ Λ W = Λ 0 −ΔΛ, Λ 0 + ΔΛ = Λ E ] × [ Φ S = Φ 0 −ΔΦ, Φ 0 + ΔΦ = Φ N ] if the Airy optimal dilatation
factor ρ 0 of type ( I.39 ), first equation, or of type ( I.39 ), second equation, is implemented.
Definition I.6 and Corollary I.7 apply accordingly. As a basis for a discussion of the Airy optimal
UPC, let us here refer to Tables I.2 and I.3 as well as to Fig. I.2 , where the Airy optimal dilatation
factor ρ 0 ( Φ 0 S = Φ 0
ΔΦ, Φ N = Φ 0 + ΔΦ ) as a function of the strip width Φ N
Φ S =2 ΔΦ
with respect to WGS 84 has been computed or plotted, respectively.
Fig. I.2. Airy optimal dilatation factor ρ 0 for a symmetric strip, generalized UPC. ρ 0 ( Φ 0 ,ΔΦ ): Airy optimal
dilatation factor ρ 0 as a function of the chosen parallel circle latitude Φ 0 and strip width ΔΦ , WGS 84. Symmetric
strip [ Λ W = Λ 0
ΔΛ, Λ 0 + ΔΛ = Λ E ]
×
[ Φ S = Φ 0
ΔΦ, Φ 0 + ΔΦ = Φ N ]
Finally, we present as an example the Airy optimal UPC for a strip system which extends
12 of southerly and +8 of northerly latitude. Again this example can be considered as
analogous to one given by Grafarend ( 1995 ) for the optimal transverse Mercator projection.
to
 
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