9-2 The conformal mappings “ellipsoid-of-revolution to plane” 265
φ
φ
φ
ln Λ
(ln Λ)
(ln Λ)
Fig. 9.1. The three postulates of the Gauss mapping “ellipsoid-of-revolution to sphere”.
Let us fix the integration constants a, c, and r. In the so-called “fundamental point” P 0 (Λ 0 , Φ 0 ), we
assume r := N (Φ 0 ), where the curvature form N 0 is defined by (9.44) (“first proposal of C. F. Gauss”).
Around the “fundamental point” P 0 (Λ 0 , Φ 0 ), we assume the Taylor expansion that is defined by (9.45)
(“second proposal of C. F. Gauss”).
r = N 0 =
A 1
1 − E 2 sin
2 Φ
,
(9.44)
ln Λ = ln Λ 0 +
d ln Λ
dφ
φ 0
(φ − φ 0 ) +
1
2
d
2 ln Λ
dφ
2
φ 0
(φ − φ 0 )
2 + · · · .
(9.45)
The following three postulates specify the above relations. Note that we can summarize the three
postulates in such a way that we postulate a horizontal turning tangent according to Fig. 9.1.
Postulate (first postulate).
Λ 0 = 1 .
(9.46)
In the so-called “fundamental point” P 0 (Λ 0 , Φ 0 ), we assume an equal lateral mapping.
ln Λ 0 = 0 .
(9.47)
End of Postulate.
Postulate (second postulate).
(ln Λ)
(φ 0 ) = 0 .
(9.48)
End of Postulate.
Postulate (third postulate).
(ln Λ)
(φ 0 ) = 0 .
(9.49)
End of Postulate.