268 9 “Ellipsoid-of-revolution to sphere and from sphere to plane”
Proof (lemma relations).
Intermediate results:
r =
N 0 cos Φ 0
a cos φ 0
,
(9.59)
a sin φ 0 = sin Φ 0 ,
(9.60)
a cos φ 0 = a
1 − sin
2 φ 0 =
1 − E 2 sin
2 Φ 0
1 − E 2
cos Φ 0 =
N 0
M 0
cos Φ 0 .
(9.61)
Action item: combine (9.59) and (9.61) and find
r =
A 1 cos Φ 0
a cos φ 0
1
1 − E 2 sin
2 Φ 0
=
A 1
√
1 − E 2
1 − E 2 sin
2 Φ 0
=
M 0 N 0 .
(9.62)
Action item: combine (9.60) and (9.61) and find
tan φ 0 =
1 − E 2
1 − E 2 sin
2 Φ 0
tan Φ 0 =
M 0
N 0
tan Φ 0 .
(9.63)
Action item: combine (9.60) and (9.61) and find
a =
sin
2 Φ 0 +
N 0
M 0
cos 2 Φ 0 = cos Φ 0
N 0
M 0
+ tan
2 Φ 0 .
(9.64)
(If N 0 /M 0 = 1, then a = 1.)
Solve the mapping equations at the initial fundamental point P 0 (Λ 0 , Φ 0 )
with respect to the integration constant c and find
c
a =
tan
π
4 +
φ 0
2
tan
a
π
4 +
Φ 0
2
1−E sin Φ 0
1+E sin Φ 0
aE/2 ,
c =
tan
π
4 +
φ 0
2
1/a
tan
π
4 +
Φ 0
2
1−E sin Φ 0
1+E sin Φ 0
E/2 .
(9.65)
End of Proof.