E-3 The Newton form of a geodesic in Maupertuis gauge (sphere, ellipsoid-of-revolution) 555
Continuation of Box.
Variant one (UC). Equidistant map of the parallel circle Φ 0 /line-of-contact:
n = sin Φ 0 , c = R
cot Φ 0
`
tan
` π
4
−
Φ
2
´´ n ,
(E.54)
λ =
cos Φ
cos Φ 0
tan
` π
4
−
Φ 0
2
´
tan
` π
4
−
Φ
2
´
! n
=
1
n
sin arctan
„
1
c
p
(q 1 ) 2 + (q 2 ) 2
« 1/n
,
(E.55)
ds
2 =
1
n 2 sin
2 arctan
„
1
c
p
(q 1 ) 2 + (q 2 ) 2
« 1/n ˆ
(dq
1 )
2 + (dq
2 )
2 ˜
.
(E.56)
Variant two (UC). Equidistant map of two parallel circles Φ 1 /Φ 2 (Lambert conformal projection):
n =
ln
cos Φ 1
cos Φ 2
ln
tan
“
π
4
−
Φ 1
2
”
tan
“
π
4
−
Φ 2
2
”
, c =
R cos Φ 1
n
`
tan
` π
4
−
Φ 1
2
´´ n ,
(E.57)
λ =
cos Φ
cos Φ 2
tan
` π
4
−
Φ 2
2
´
tan
` π
4
−
Φ
2
´
! n
=
1
n
sin arctan
„
1
c
p
(q 1 ) 2 + (q 2 ) 2
« 1/n
,
(E.58)
ds
2 =
1
n 2 sin
2 arctan
„
1
c
p
(q 1 ) 2 + (q 2 ) 2
« 1/n ˆ
(dq
1 )
2 + (dq
2 )
2 ˜
.
(E.59)
Box E.2 (The Lagrangean version versus the Hamiltonian version of a geodesic in S
2
R in terms of conformal
coordinates (isometric coordinates) and Maupertuis gauge).
(i) Universal Polar Stereographic Projection (UPS):
L
2 (q(t), ˙
q(t)) =
1
2
δ µν ˙
q
µ ˙
q
ν +
8R
4
(4R 2 + (q 1 ) 2 + (q 2 ) 2 ) 2 ,
H
2 (q(t), p(t)) =
1
2
(p
2
1 + p
2
2 ) −
8R
4
(4R 2 + (q 1 ) 2 + (q 2 ) 2 ) 2 .
(E.60)
(ii) Universal Mercator Projection (UM):
L
2 (q(t), ˙
q(t)) =
1
2
δ µν ˙
q
µ ˙
q
ν +
1
2 cosh
2 (q 2 /R)
, H
2 (q(t), p(t)) =
1
2
(p
2
1 + p
2
2 ) −
1
2 cosh
2 (q 2 /R)
. (E.61)
(iii) Universal Transverse Mercator Projection (UTM):
L
2 (q(t), ˙
q(t)) =
1
2
δ µν ˙
q
µ ˙
q
ν +
1
2
cos
2 arcsin[tanh(q
2 /R)] ,
H
2 (q(t), p(t)) =
1
2
(p
2
1 + p
2
2 ) −
1
2
cos
2 arcsin[tanh(q
2 /R)] .
(E.62)
(iv) Universal Conic Projection (UC):
L
2 (q(t), ˙
q(t)) =
1
2
δ µν ˙
q
µ ˙
q
ν +
1
2n
sin
2 arctan
„
1
c
p
(q 1 ) 2 + (q 2 ) 2
« 1/n
,
H
2 (q(t), p(t)) =
1
2
(p
2
1 + p
2
2 ) −
1
2n
sin
2 arctan
„
1
c
p
(q 1 ) 2 + (q 2 ) 2
« 1/n
.
(E.63)