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(1 +
A
2
A
3
Λ
)
2
(1 +
A
1
A
3
Λ
)
2
(1 +
A
1
A
2
Λ
)
2
A
1
A
2
A
3
=0
or
a
6
(
Λ
2
)
3
+
a
4
(
Λ
2
)
2
+
a
2
(
Λ
2
)+
a
0
=0
−
(J.23)
(cubic equation for
Λ
)
.
Backward step:
Λ
into (i), (ii), and (iii), and solve the three equations for
x
1
, x
2
.and
x
3
.and
test the condition of positivity of the Hesse matrix in order to discriminate the admissible
solutions.”
“Reset
x
4
∼
J-2 Gauss Surface Normal Coordinates: Case Study
Ellipsoid-of-Revolution
First, we review surface normal coordinates for the ellipsoid-of-revolution. Second, we extend the
derivation to three-dimensional surface normal coordinates in terms of the forward transformation
as well as of the backward transformation by means of the constraint minimum distance mapping
in terms of the Buchberger algorithm.
J-21 Review of Surface Normal Coordinates
for the Ellipsoid-of-Revolution
The coordinates of the ellipsoid-of-revolution of type
{
ellipsoidal longitude, ellipsoidal latitude,
are
surface normal coordinates
in the following sense. They are founded on
the famous
Gauss map
of the surface normal vector
ellipsoidal height
}
) of the ellipsoid-of-revolution (
J.24
)in
terms of the semimajor axis
A
1
>A
2
and of the semi-minor axis
A
2
<A
1
.
ν
(
x
ν
(
x
) is also called
normal field
. Compare with Fig.
J.5
.
:=
x
3
+
.
f
(
x, y, z
):=
x
2
+
y
2
A
1
+
z
2
2
A
1
,A
2
+
E
∈
R
A
2
−
1=0
,
R
A
1
>A
2
∈
R
(J.24)
Definition J.1 (Gauss map).
2
:=
3
x
2
+
y
2
+
z
2
The spherical image of the surface norm vector
ν
(
l,b
)
∈
S
{
x
∈
R
|
−
1=0
}
of
2
3
is defined as (
J.25
) with respect to the orthonormal left
the ellipsoid-of-revolution
E
A
1
,A
2
⊂
R
basis
{
e
1
,
e
2
,
e
3
|O}
in the origin
O
of the ellipsoid-of-revolution.
ν
(
l, b
)=
e
1
cos
b
cos
l
+
e
2
cos
b
sin
l
+
e
3
sin
b
=
⎡
⎤
cos
b
cos
l
cos
b
sin
l
sin
b
⎣
⎦
∈
2
A
1
,A
2
=[
e
1
,
e
2
,
e
3
]
N
lb
E
.
(J.25)
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