Geography Reference
In-Depth Information
point
P ∈
T
to
p ∈
E
A
1
,A
2
. Finally, by means of Box
J.5
, we convert the Cartesian coordinates
{X, Y, Z}∈
T
to Gauss ellipsoidal coordinates
{L, B, H}
.
Without the various forward and backward reduction steps, we could automatically generate an
equivalent algorithm for solving the normal equations in a closed form by means of
Grobner basis
and the
Buchberger algorithm
(
Cox et al. 1996
;
Becker and Weispfennig 1998
;
Sturmfels 1996
;
Zippel 1993
). Let us write the
Ideal
of the polynomials in lexicographic order
x
1
>x
2
>x
3
>x
4
(read:
x
1
before
x
2
before
x
3
before
x
4
)intoBox
J.6
.TheGrobner basis of the
Ideal
characteristic
for the minimum distance mapping problem can be computed either by MATHEMATICA software
or by MAPLE software.
2
and
{x
1
,x
2
,x
3
}∈
E
A
1
,A
2
Box J.6 (Algorithm for solving the normal equations of the constraint minimum distance
mapping).
First forward step. Solve (i)
,
(ii)
,
and (iii) for
{
x
1
,x
2
,x
3
}
:
(i)
X
1+
b
2
x
4
,
x
1
(1 +
b
2
x
4
)=
X ⇒ x
1
=
(ii)
(J.50)
Y
1+
b
2
x
4
,
x
2
(1 +
b
2
x
4
)=
Y
x
2
=
⇒
(iii)
Z
1+
a
2
x
4
.
x
3
(1 +
a
2
x
4
)=
Z
x
3
=
⇒
x
1
,x
2
,x
3
,x
4
}
Second forward step. Substitute
{
:
1
1
x
∧
1
+
x
∧
2
=
(1 +
b
2
x
4
)
2
(
X
2
+
Y
2
)
,
∧
3
=
(1 +
a
2
x
4
)
2
Z
2
,
(J.51)
⇔ b
2
X
2
+
Y
2
(1 +
b
2
x
4
)
2
b
2
(
x
∧
1
+
x
∧
2
)+
a
2
x
∧
2
− a
2
b
2
3
Z
2
(1 +
a
2
x
4
)
2
−
+
a
2
a
2
b
2
=0
.
The characteristic quadratic equation.
Multiply the rational equation of constraint by (1 +
a
2
x
4
)
2
(1 +
b
2
x
4
)
2
:
b
2
(1 +
a
2
x
4
)
2
(
X
2
+
Y
2
)+
a
2
(1 +
b
2
x
4
)
2
Z
2
−
a
2
b
2
(1 +
a
2
x
4
)
2
(1 +
b
2
x
4
)
2
=0
⇔
(J.52)
(1 + 2
a
2
x
4
+
a
4
x
4
)
b
2
(
X
2
+
Y
2
)+(1+2
b
2
x
4
+
b
4
x
4
)
a
2
Z
2
−
a
2
b
2
(1 + 2
a
2
x
4
+
a
4
x
4
)(1 + 2
b
2
x
4
+
b
4
x
4
)=0
,
−
−x
4
a
6
b
6
−
2
x
4
a
4
b
4
(
a
2
+
b
2
)+
x
4
a
2
b
2
[
a
2
(
X
2
+
Y
2
)+
b
2
Z
2
−
4
a
2
b
2
− a
4
b
4
]+
−
(J.53)
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