632 I Mercator projection and polycylindric projection
Let us refer to the representation of the areal elements {dS, S} of the ellipsoid-of-revolution of
Section I-1. With the next step, we move on to Lemma I.11 for a representation of J lA subject to
Λ 1 = Λ 2 , in particular (I.34).
Lemma I.11 (Generalized polycylindric projection, Airy distortion energy).
In case of the generalized polycylindric projection of conformal type, the left Airy distortion energy
J lA is the quadratic form in terms of the dilatation factor ρ 0 , in particular
J lA (ρ 0 ) = c 00 − 2c 01 ρ 0 + c 02 ρ
2
0 ,
(I.37)
such that
c 00 = 1 ,
c 01 = c 1 cos Φ 0 /
1 − E 2 sin
2 Φ 0 ,
c 02 = c 2 cos
2 Φ 0 /(1 − E
2 sin
2 Φ 0 )
(I.38)
hold.
End of Lemma.
Once we start from the proof of Lemma I.4, the extension to the result of (I.37) and I.38) with
respect to (I.34) into Lemma I.11 is straightforward.
Question.
Question: “Where, with respect to the dilatation factor ρ 0 , is the Airy distortion energy
minimal?” Answer: “The detailed answer is given in Lemma I.12.”
Lemma I.12 (Minimal Airy distortion energy).
The Airy distortion energy (I.37) is minimal if the dilatation factor amounts to ˆ
ρ 0 = c 01 /c 02 and
ˆ
ρ 0 (Φ S , Φ N ) =
= ˆ
ρ 0 (Φ S , Φ N )
1 − E 2 sin
2 Φ 0
cos Φ 0
,
ˆ
ρ 0 (Φ S = −Φ N ) =
= ˆ
ρ 0 (Φ S = −Φ N )
1 − E 2 sin
2 Φ 0
cos Φ 0
,
(I.39)
where c 01 and c 02 follow from (I.38) and (I.10), ˆ
ρ 0 (Φ S = Φ N ) from (I.19) (see the first equation), and
ˆ
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 − ∆Λ, Λ 0 + ∆Λ = Λ E ] × [Φ S = Φ 0 − ∆Φ, Φ 0 + ∆Φ = Φ N ] if the Airy optimal dilatation factor
ˆ
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 Table I.2 and Table I.3 as well as to Fig. I.2, where the Airy optimal dilatation factor
ˆ
ρ 0 (Φ 0 , Φ S = Φ 0 − ∆Φ, Φ N = Φ 0 + ∆Φ) as a function of the strip width Φ N − Φ S = 2∆Φ with respect
to WGS 84 has been computed or plotted, respectively.
Let us refer to the representation of the areal elements {dS, S} of the ellipsoid-of-revolution of
Section I-1. With the next step, we move on to Lemma I.11 for a representation of J lA subject to
Λ 1 = Λ 2 , in particular (I.34).
Lemma I.11 (Generalized polycylindric projection, Airy distortion energy).
In case of the generalized polycylindric projection of conformal type, the left Airy distortion energy
J lA is the quadratic form in terms of the dilatation factor ρ 0 , in particular
J lA (ρ 0 ) = c 00 − 2c 01 ρ 0 + c 02 ρ
2
0 ,
(I.37)
such that
c 00 = 1 ,
c 01 = c 1 cos Φ 0 /
1 − E 2 sin
2 Φ 0 ,
c 02 = c 2 cos
2 Φ 0 /(1 − E
2 sin
2 Φ 0 )
(I.38)
hold.
End of Lemma.
Once we start from the proof of Lemma I.4, the extension to the result of (I.37) and I.38) with
respect to (I.34) into Lemma I.11 is straightforward.
Question.
Question: “Where, with respect to the dilatation factor ρ 0 , is the Airy distortion energy
minimal?” Answer: “The detailed answer is given in Lemma I.12.”
Lemma I.12 (Minimal Airy distortion energy).
The Airy distortion energy (I.37) is minimal if the dilatation factor amounts to ˆ
ρ 0 = c 01 /c 02 and
ˆ
ρ 0 (Φ S , Φ N ) =
= ˆ
ρ 0 (Φ S , Φ N )
1 − E 2 sin
2 Φ 0
cos Φ 0
,
ˆ
ρ 0 (Φ S = −Φ N ) =
= ˆ
ρ 0 (Φ S = −Φ N )
1 − E 2 sin
2 Φ 0
cos Φ 0
,
(I.39)
where c 01 and c 02 follow from (I.38) and (I.10), ˆ
ρ 0 (Φ S = Φ N ) from (I.19) (see the first equation), and
ˆ
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 − ∆Λ, Λ 0 + ∆Λ = Λ E ] × [Φ S = Φ 0 − ∆Φ, Φ 0 + ∆Φ = Φ N ] if the Airy optimal dilatation factor
ˆ
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 Table I.2 and Table I.3 as well as to Fig. I.2, where the Airy optimal dilatation factor
ˆ
ρ 0 (Φ 0 , Φ S = Φ 0 − ∆Φ, Φ N = Φ 0 + ∆Φ) as a function of the strip width Φ N − Φ S = 2∆Φ with respect
to WGS 84 has been computed or plotted, respectively.