Ellipsoid-of-Revolution to Sphere and from Sphere to Plane - Map Projections: Cartographic Information Systems - page 296

(“spherical isometric latitude”) .

Second standard integral:

E 2

1 − d Φ

cos Φ = I second .

(9.40)

E 2 sin 2 Φ

1 −

E cos Φ
1+ E sin Φ −

,

1 − E 2 1

cos Φ =

1
cos Φ −

E
2

E cos Φ

(9.41)

E 2 sin 2 Φ 1 − 1 − E sin Φ

1 − E 2 d Φ

cos Φ =

E 2 sin 2 Φ

1 −

=lntan π

4 + Φ

E

2 ln 1+ E sin Φ

−

E sin Φ =

2 1 −

ln tan π

E/ 2 := Q

1

4 + Φ

E sin Φ
1+ E sin Φ

−

(9.42)

2

(“ellipsoidal isometric latitude”) .

The combination of the first and the second standard integral leads us to the celebrated relation
in terms of the integration constants c and k ,namely

q = a ( Q + k ) ,k =ln c, c =exp k ;

ln tan π

= a lntan π

4 + φ 4 + Φ

aE
2

ln 1+ E sin Φ −

1 − E sin Φ + a ln c.

(9.43)

2 2

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”).

A 1

r = N 0 = 1 , (9.44)

E 2 sin 2 Φ −

ln Λ =ln Λ 0 dln Λ
d φ

d 2 ln Λ
d φ 2

( φ − φ 0 )+ 1
2

( φ − φ 0 ) 2 + ··· .

(9.45)

φ 0 φ 0

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 .