“Sphere to Cylinder”: Polar Aspect - Map Projections: Cartographic Information Systems - page 312

Papers are Airy ( 1861 ), Francula ( 1971 ), Grafarend ( 1995 ), Grafarend ( 1984 ), Grafarend and
Syffus ( 1998c ), Hojovec and Jokl ( 1981 ), Jordan ( 1875 , 1896 ), Kaltsikis ( 1980 ), Kavrajski ( 1958 ).

Fig. 10.8. The Airy-Kavrajski optimum of three different mappings: (i) conformal maps, (ii) equiareal maps,
and (iii) distance preserving maps

Let us finally prove our statements based upon (i) the various mapping equations of the sphere
under the postulates of equidistant mappings on two parallels and type cylinder mappings and
(ii) the corresponding principal stretches. The Airy distortion energy is based upon the inte-
gral ( 10.15 ), the global arithmetic mean of the surface integral of a spherical zone between the
equator and the latitude circle Φ of the local measure respective global measure ( 10.16 ).

1
2 S

1) 2 +( Λ 2 1) 2 ] , I A := d S [( Λ 1 − − (10.15)

S

d S =2 πR 2 cos Φ d Φ versus S =2 πR 2 sin Φ. (10.16)

Proof (conformal cylindrical mapping).

We start from the mapping equations of conformal type constrained to the equidistance postulate
on two parallel circles. In addition, we enjoy the identity postulate of left principal stretches.
x
y

= R cos Φ 0 Λ = R cos Φ 0 Λ ,

(10.17)

ln cot( 4 −

Φ

ln tan( 4 + 2 )

2 )

Λ 1 = Λ 2 = cos Φ 0

cos Φ .

(10.18)

Starting from the above relations, we obtain