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z
(
l,b
)=
A
1
(1
−
E
2
)sin
b
1
.
E
2
sin
2
b
−
End of Lemma.
We have the proof that the normal field
(
l, b
) is not a gradient field, in consequence an anholo-
nomic variable, to the reader. In terms of surface normal coordinates, the differential invariants
{
ν
I, II, III
}
take a simple form, namely
2
:=
I
∼
d
μ
μ
l
d
l
+
μ
b
d
b
|
μ
l
d
l
+
μ
b
d
b
,
(J.42)
II
∼−
d
μ
|
d
ν
:=
μ
l
d
l
+
μ
b
d
b
|
ν
l
d
t
+
ν
b
d
b
,
(J.43)
2
:=
III
∼
d
ν
ν
l
d
l
+
ν
b
d
b
|
ν
l
d
l
+
ν
b
d
b
,
(J.44)
or
A
1
cos
2
b
A
1
(1
E
2
)
2
−
2
=
1
− E
2
sin
2
b
d
l
2
+
(1
− E
2
sin
2
b
)
3
/
2
d
b
2
,
I
∼
d
μ
(J.45)
A
1
cos
2
b
E
2
)
A
1
(1
−
1
d
l
2
+
(1
− E
2
sin
2
b
)
3
/
2
d
b
2
,
II
∼
d
μ
|
d
ν
=
(J.46)
E
2
sin
2
b
−
2
=cos
2
b
d
l
2
+d
b
2
.
III ∼
d
ν
(J.47)
2
A
1
,A
2
Corollary J.4 (
E
Gauss differential invariants).
The Gauss differential invariants
{I,II,III}
of the ellipsoid-of-revolution
E
A
1
,A
2
are characterized
by Gauss surface normal coordinates represented by (
J.42
), (
J.43
), and (
J.44
). Especially, the
Gauss map
N
A
1
,A
2
2
has the spherical metric
III
.
E
→
S
End of Corollary.
J-22 Buchberger Algorithm of Forming a Constraint Minimum
Distance Mapping
The forward transformation of Gauss coordinates of an ellipsoid-of-revolution takes the form (
J.48
)
and (
J.49
) illustrated by Fig.
J.6
. The triplet
{
surface normal longitude, surface normal latitude,
surface normal height
}
describes the position of a point
X
(
L, B, H
) where the surface normal
height
H
(
L, B
) is a given function of longitude and latitude.
Box J.5 (Forward transformation of Gauss coordinates of an ellipsoid-of-revolution).
X
(
L, B, H
)=+
e
1
+
H
(
L, B
)
cos
B
cos
L
A
1
1
E
2
sin
2
B
−
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