1-10 Two examples: Mercator Projection and Stereographic Projection 69
Continuation of Box.
Left differential equation of the factor of conformality (E
2
A 1 ,A 1 ,A 2 ):
k l =
(1 − E
2 sin
2 φ)
2
A
2
1 (1 − E 2 )
=
=
1 − E
2 sin
2 f
−1
“ p
P 2 + Q 2
”
A
2
1 (1 − E 2 )
,
∆ ln λ
2
l + 2k(P, Q)λ
2
l = 0 ,
λ
2
l =
A
2
1 cos
2 f
−1
“ p
P 2 + Q 2
”
1 − E 2 sin
2 f −1
“ p
P 2 + Q 2
”
1
P 2 + Q 2 ,
ln λ
2
l = ln A
2
1 + 2 ln cos f
−1
“ p
P 2 + Q 2
”
− ln
h
1 − E
2 sin
2 f
−1
“ p
P 2 + Q 2
”i
− ln
`
P
2 + Q
2 ´
,
∆ l ln λ
2
l =
`
D
2
P + D
2
Q
´
ln λ
2
l = −2k(P, Q)λ
2
l
q. e. d.
(1.255)
Box 1.37 (Representation of Φ
−1
l
and Φ
−1
r
in terms of conformal coordinates: E
2
A 1 ,A 1 ,A 2 → S
2
r ).
Φ
−1
l
: X (Λ, Φ) =
Φ
−1
r
: x(λ, φ) =
= E 1
A 1 cos Φ cos Λ
p
1 − E 2 sin
2 Φ
+
= e 1 r cos φ cos λ +
+ E 2
A 1 cos Φ sin Λ
p
1 − E 2 sin
2 Φ
+
+ e 2 r cos φ sin λ +
+ E 3
A 1 (1 − E
2 ) sin Φ
p
1 − E 2 sin
2 Φ
=
+
e 3 r sin φ
=
= E 1
A 1 cos f
−1
“ p
P 2 + Q 2
”
r
1 − E 2 sin
2 f −1
“ p
P 2 + Q 2
”
P
p
P 2 + Q 2
+
= e 1 4r
2
p
4r 2 +p 2 +q 2
+
+ E 2
A 1 cos f
−1
“ p
P 2 + Q 2
”
r
1 − E 2 sin
2 f −1
“ p
P 2 + Q 2
”
Q
p
P 2 + Q 2
+
+e 2 4r
2
q
4r 2 + p 2 + q 2 +
+
E 3
A 1 (1 − E
2 ) sin f
−1
“ p
P 2 + Q 2
”
r
1 − E 2 sin
2 f −1
“ p
P 2 + Q 2
” .
+ e 3 r
4r
2 −(p
2 +q
2 )
4r 2 +p 2 +q 2 .
(1.256)
Isoparametric mapping:
p = P , q = Q .
(1.257)