Mercator Projection and Polycylindric Projection - Map Projections: Cartographic Information Systems - page 816

matrix of the right metric tensor of the plane generated by developing the circular cylinder C

2 R

of radius ( I.31 ), namely the unit matrix G r =I 2 .

End of Lemma.

Definition I.10 (Generalized polycylindric projection, Airy optimum).

The generalized polycylindric projection of conformal type of ( I.33 )and( I.34 ) of the ellipsoid-of-
revolution E

2 R of radius ( I.31 ) is called Airy optimal if
the deviation from an isometry ( I.35 ) in terms of the left principal stretches {Λ 1 ,Λ 2 } , in particu-
lar ( I.34 ), averaged over a mapping area of interest, namely the surface integral ( I.36 ), is minimal
with respect to the unknown dilatation factor ρ 0 .

A 1 ,A 2 onto the developed circular cylinder C

1) 2 +( Λ 2 − 1) 2 ] / 2 , [( Λ 1 − (I.35)

1
2 S

[( Λ 1 − 1) 2 +( Λ 2 − 1) 2 ]d S =min
ρ 0

J l A := .

(I.36)

area

End of Definition.

Let us refer to the representation of the areal elements { d S, S} of the ellipsoid-of-revolution
of Sect. I-1 . With the next step, we move on to Lemma I.11 for a representation of J l A 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 l A is the quadratic form in terms of the dilatation factor ρ 0 , in particular

2 c 01 ρ 0 + c 02 ρ 0 ,

J l A ( ρ 0 )= c 00 − (I.37)

such that
c 00 =1 ,
c 01 = c 1 cos Φ 0 / 1 E 2 sin 2 Φ 0 ,

− (I.38)

c 02 = c 2 cos 2 Φ 0 / (1 E 2 sin 2 Φ 0 ) −

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: “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 .”