Geography Reference
In-Depth Information
Fig. 24.1.
Angle and distance ration in
E
3
in a triangle spanned by
X
0
,
X
1
,
X
2
Theorem 24.1 (Jacobi map, equivariance of angles and distance ratios with respect to
C
(3) close
to identity).
d
Y
=
J
dX
of type
C
(3), formulae
(
24.17
)-(
24.19
)
leaves
angle Ψ and distance ratios Ω of type formula (
24.16
), (
24.17
) equivariant.
The conformal Jacobi map d
X
→
End of Theorem.
d
Y
1
d
Y
2
d
X
1
J
1
J
2
d
X
2
cos
Ψ
y
:=
d
Y
1
/d
Y
2
d
X
1
J
1
J
1
d
X
1
X
2
J
2
J
2
d
X
2
J
1
J
2
=(
I
+
δ
A
+
I
δc
)
T
(
I
+
δ
A
+
I
δc
)=
I
+
δ
A
T
+
I
δc
+
δ
A
+
I
δc
+
o
(
δc
2
)
δ
A
T
=
d
Y
1
d
Y
1
d
Y
2
d
Y
2
=
=
d
Y
1
d
Y
2
→
(24.20)
−
δ
A
J
1
J
2
=
I
(1 + 2
δc
)
,
J
1
J
1
=
J
2
J
2
=
I
(1 + 2
δc
)
(24.21)
(1 + 2
δc
)
d
X
1
d
X
2
d
X
1
d
X
2
cos
Ψ
y
=
(1 + 2
δc
)
d
X
1
d
X
1
(1 + 2
δc
)
X
2
d
X
2
=
d
X
1
d
X
1
X
2
d
X
2
=cos
Ψ
x
Ω
y
=
d
Y
1
d
Y
1
=
d
X
1
J
1
J
1
d
X
1
Ω
y
=
d
X
1
d
X
1
X
2
d
X
2
X
2
J
2
J
2
d
X
2
formula
(
24.20
)
d
Y
2
d
Y
2
→
=
Ω
x
This contribution is taken from
Grafarend and Kampmann
(
1996
).
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