628 I Mercator projection and polycylindric projection
Proof.
ˆ
ρ = c 1 /c 2 is proven by the following procedure.
J lA = c 0 − 2c 1 ρ + c 2 ρ
2 = min
ρ
.
(I.20)
Necessary:
dJ lA
dρ
(ρ = ˆ
ρ) = −2c 1 + c 2 ˆ
ρ = 0
⇔
ˆ
ρ = c 1 /c 2 .
(I.21)
Sufficient:
d
2 J lA
dρ 2 (ρ = ˆ
ρ) = 2c 2 > 0 .
(I.22)
(I.19) directly follows from (I.10) and ˆ
ρ = c 1 /c 2 .
End of Proof.
Before we go into numerical computations of the optimal dilatation factor ˆ
ρ for the generalized Mercator projection, let us here briefly present a result for zero total areal distortion as it is outlined by
E. Grafarend (1995).
Definition I.6 (Generalized Mercator projection, optimal with respect to areal distortion).
The generalized Mercator projection of the ellipsoid-of-revolution E
2
A 1 ,A 2
onto the developed circular
cylinder C
2
ρA 1
of radius ρA 1 is called optimal with respect to areal distortion if the deviation from an
equiareal mapping Λ 1 Λ 2 − 1 in terms of the left principal stretches (Λ 1 , Λ 2 ) averaged over a mapping
area of interest, namely the total areal distortion (I.23), is minimal with respect to the unknown
dilatation factor ρ.
J l :=
1
S
S
(Λ 1 Λ 2 − 1)dS = min
ρ
.
(I.23)
End of Definition.
Corollary I.7 (Generalized Mercator projection, dilatation factor).
For a generalized Mercator projection of the half-symmetric strip [Λ W = Λ 0 − ∆Λ, Λ 0 + ∆Λ =
Λ E ] × [Φ S , Φ N ], the postulates of minimal Airy distortion energy (minimal total distance distortion)
and of minimal total areal distortion lead to the same unknown dilatation factor ˆ
ρ of (I.19) by firstorder approximation. The total areal distortion amounts to zero.
End of Corollary.
Proof.
We start from the representation of the left principal stretches for a mapping of conformal type
implemented into, firstly, J lA , secondly, J l . The squared left principal stretches are assumed to be
given by 1 plus a small quantity µ except for the dilatation factor ρ:
Λ
2
1 = Λ
2
2 = Λ
2
S = ρ
2 (1 + µ) ∀ µ 1 ,
Λ 1 = Λ 2 = Λ S = ρ
1 +
µ
2
+ O(µ
2 ) ∀ µ 1 .
(I.24)
Proof.
ˆ
ρ = c 1 /c 2 is proven by the following procedure.
J lA = c 0 − 2c 1 ρ + c 2 ρ
2 = min
ρ
.
(I.20)
Necessary:
dJ lA
dρ
(ρ = ˆ
ρ) = −2c 1 + c 2 ˆ
ρ = 0
⇔
ˆ
ρ = c 1 /c 2 .
(I.21)
Sufficient:
d
2 J lA
dρ 2 (ρ = ˆ
ρ) = 2c 2 > 0 .
(I.22)
(I.19) directly follows from (I.10) and ˆ
ρ = c 1 /c 2 .
End of Proof.
Before we go into numerical computations of the optimal dilatation factor ˆ
ρ for the generalized Mercator projection, let us here briefly present a result for zero total areal distortion as it is outlined by
E. Grafarend (1995).
Definition I.6 (Generalized Mercator projection, optimal with respect to areal distortion).
The generalized Mercator projection of the ellipsoid-of-revolution E
2
A 1 ,A 2
onto the developed circular
cylinder C
2
ρA 1
of radius ρA 1 is called optimal with respect to areal distortion if the deviation from an
equiareal mapping Λ 1 Λ 2 − 1 in terms of the left principal stretches (Λ 1 , Λ 2 ) averaged over a mapping
area of interest, namely the total areal distortion (I.23), is minimal with respect to the unknown
dilatation factor ρ.
J l :=
1
S
S
(Λ 1 Λ 2 − 1)dS = min
ρ
.
(I.23)
End of Definition.
Corollary I.7 (Generalized Mercator projection, dilatation factor).
For a generalized Mercator projection of the half-symmetric strip [Λ W = Λ 0 − ∆Λ, Λ 0 + ∆Λ =
Λ E ] × [Φ S , Φ N ], the postulates of minimal Airy distortion energy (minimal total distance distortion)
and of minimal total areal distortion lead to the same unknown dilatation factor ˆ
ρ of (I.19) by firstorder approximation. The total areal distortion amounts to zero.
End of Corollary.
Proof.
We start from the representation of the left principal stretches for a mapping of conformal type
implemented into, firstly, J lA , secondly, J l . The squared left principal stretches are assumed to be
given by 1 plus a small quantity µ except for the dilatation factor ρ:
Λ
2
1 = Λ
2
2 = Λ
2
S = ρ
2 (1 + µ) ∀ µ 1 ,
Λ 1 = Λ 2 = Λ S = ρ
1 +
µ
2
+ O(µ
2 ) ∀ µ 1 .
(I.24)