646 J Gauss surface normal coordinates in geometry and gravity space
ν(l, b) = ν(x, y, z) ,
(J.32)
⎡
⎢
⎣
cos b cos l
cos b sin l
sin b
⎤
⎥
⎦ =
⎡
⎢
⎣
x
y
z/(1 − E
2 )
⎤
⎥
⎦
1 − E
2
(1 − E 2 ) 2 (x 2 + y 2 ) + z 2
,
(J.33)
f (x, y, z) =
x
2 + y
2
A 2
1
+
z
2
A 2
2
− 1 or x
2 + y
2 +
z
2
1 − E 2 = A
2
1 .
(J.34)
Box J.4 (Four steps towards an isometric embedding {x(l, b), y(l, b), z(l, b)} of E
2
A 1 ,A 2 ⊂ R
3 ).
The first operation. Use (J.33), the first and second equation, and add
cos
2 b = (1 − E
2 )
2
x
2 + y
2
(1 − E 2 ) 2 (x 2 + y 2 ) + z 2
⇒
x
2 + y
2 =
ˆ
x
2 + y
2 + z
2 /(1 − E
2 )
2
˜
cos
2 b .
(J.35)
The second operation. Use (J.33), the third equation:
sin
2 b =
z
2
(1 − E 2 ) 2 (x 2 + y 2 ) + z 2
⇒
z
2 /(1 − E
2 ) = (1 − E
2 )
ˆ
x
2 + y
2 + z
2 /(1 − E
2 )
2 ˜
sin
2 b .
(J.36)
The third operation. Replace the terms in (J.34) by (J.35) and (J.36):
A
2
1 = x
2 + y
2 + z
2 /(1 − E
2 ) =
ˆ
x
2 + y
2 + z
2 /(1 − E
2 )
2 ˜
(1 − E
2 sin
2 b)
⇒
x
2 + y
2 + z
2 /(1 − E
2 )
2 =
A
2
1
1 − E 2 sin
2 b
⇒
p
x 2 + y 2 + z 2 /(1 − E 2 ) 2 = A 1 /
p
1 − E 2 sin
2 b .
(J.37)
The fourth operation. Solve (J.33) for {x, y, z} and replace the square root by (J.37):
x =
p
x 2 + y 2 + z 2 /(1 − E 2 ) 2 cos b cos l ,
y =
p
x 2 + y 2 + z 2 /(1 − E 2 ) 2 cos b sin l ,
z = (1 − E
2 )
p
x 2 + y 2 + z 2 /(1 − E 2 ) 2 sin b .
(J.38)
The result is summarized in Lemma J.3.