15-4 Principal distortions and various optimal designs (UTM mappings) 333
For the proof, we start from the formula Λ
2 (l, b) as a representation of formula (15.97), namely the
principal distortion as a function of the longitudinal difference L − L 0 =: l and the latitude B. The
criterion of optimality for the first design of the transverse Mercator projection modulo an unknown
dilatation factor ρ is the minimal total distance distortion over a meridian strip [l W , l E ] × [B S , B N ] between a longitudinal extension L W and L E and a latitudinal extension B S and B N (namely the symbols
S, N, E, and W as indices denote South, North, East, and West), in particular, the G. B. Airy (1861)
distortion measure (15.103) with respect to the principal distortions Λ 1 and Λ 2 and the spheroidal
surface element, locally (15.104) and globally (15.105). The G. B. Airy distortion minimization subject
to Λ 1 = Λ 2 = Λ, the criterion for conformality, leads directly to the representations (15.98)–(15.100).
I lA :=
:=
1
2S
dS
(Λ 1 − 1)
2 + (Λ 2 − 1)
2
= min ,
(15.103)
dS =
= A
2
1 (1 − E
2 ) cos B dl dB(1 − E
2 sin
2 B)
2 ,
(15.104)
S =
= 2A
2
1 (1 − E
2 )l E ×
×[sin B N + (2/3)E
2 sin
3 B N − (sin B S + (2/3)E
2 sin
3 B S )+
+O N (E
4 ) + O S (E
4 )] .
(15.105)
As an alternative, we could base a second design of the transverse Mercator projection modulo an
unknown dilatation factor ρ on a minimal total areal distortion over a meridian strip [l W , l E ]×[B S , B N ]
between a longitudinal extension L W and L E and a latitudinal extension B S and B N , in particular,
the distortion measure (15.106) with respect to principal distortions Λ 1 and Λ 2 , and the spheroidal
surface element. But surprisingly, it does not differ from the one of the previous corollary, the optimal
Airy design, see Corollary 15.8, the proof of which we want to leave to the reader.
I l (areal) :=
:=
1
S
dS
Λ 1 Λ 2 − 1
.
(15.106)
Corollary 15.8 (The dilatation factor for an optimal universal transverse Mercator projection, the zero
total distortion).
For a conformal map of the half-symmetric strip [−l E , l E ] × [B S , B N ] of type universal transverse Mercator projection, the postulate of minimal total distance distortion (Airy optimum) and the postulate
of minimal total areal distortion lead to the same unknown dilatation factor ρ (→ (15.98)–(15.100)).
The total areal distortion amounts to zero!
End of Corollary.
For the proof, we start from the formula Λ
2 (l, b) as a representation of formula (15.97), namely the
principal distortion as a function of the longitudinal difference L − L 0 =: l and the latitude B. The
criterion of optimality for the first design of the transverse Mercator projection modulo an unknown
dilatation factor ρ is the minimal total distance distortion over a meridian strip [l W , l E ] × [B S , B N ] between a longitudinal extension L W and L E and a latitudinal extension B S and B N (namely the symbols
S, N, E, and W as indices denote South, North, East, and West), in particular, the G. B. Airy (1861)
distortion measure (15.103) with respect to the principal distortions Λ 1 and Λ 2 and the spheroidal
surface element, locally (15.104) and globally (15.105). The G. B. Airy distortion minimization subject
to Λ 1 = Λ 2 = Λ, the criterion for conformality, leads directly to the representations (15.98)–(15.100).
I lA :=
:=
1
2S
dS
(Λ 1 − 1)
2 + (Λ 2 − 1)
2
= min ,
(15.103)
dS =
= A
2
1 (1 − E
2 ) cos B dl dB(1 − E
2 sin
2 B)
2 ,
(15.104)
S =
= 2A
2
1 (1 − E
2 )l E ×
×[sin B N + (2/3)E
2 sin
3 B N − (sin B S + (2/3)E
2 sin
3 B S )+
+O N (E
4 ) + O S (E
4 )] .
(15.105)
As an alternative, we could base a second design of the transverse Mercator projection modulo an
unknown dilatation factor ρ on a minimal total areal distortion over a meridian strip [l W , l E ]×[B S , B N ]
between a longitudinal extension L W and L E and a latitudinal extension B S and B N , in particular,
the distortion measure (15.106) with respect to principal distortions Λ 1 and Λ 2 , and the spheroidal
surface element. But surprisingly, it does not differ from the one of the previous corollary, the optimal
Airy design, see Corollary 15.8, the proof of which we want to leave to the reader.
I l (areal) :=
:=
1
S
dS
Λ 1 Λ 2 − 1
.
(15.106)
Corollary 15.8 (The dilatation factor for an optimal universal transverse Mercator projection, the zero
total distortion).
For a conformal map of the half-symmetric strip [−l E , l E ] × [B S , B N ] of type universal transverse Mercator projection, the postulate of minimal total distance distortion (Airy optimum) and the postulate
of minimal total areal distortion lead to the same unknown dilatation factor ρ (→ (15.98)–(15.100)).
The total areal distortion amounts to zero!
End of Corollary.