280 10 “Sphere to cylinder”: polar aspect
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
0.5
1
1.5
2
2.5
Φ
0
√I
AK
Min = 42°
Conformal and Equal−area mapping
Equidistant mapping
˛
˛
˛
˛
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 integral (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).
I A :=
1
2S
S
dS[(Λ 1 − 1)
2 + (Λ 2 − 1)
2 ] ,
(10.15)
dS = 2πR
2 cos Φ dΦ versus S = 2πR
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
ln cot
π
4 −
Φ
2
= R cos Φ 0
Λ
ln tan
π
4 +
Φ
2
,
(10.17)
Λ 1 = Λ 2 =
cos Φ 0
cos Φ
.
(10.18)