Geography Reference
In-Depth Information
15-4 Principal Distortions and Various Optimal Designs (UTM
Mappings)
Principal distortions and various optimal designs of the Universal Transverse Mercator Pro-
jection (UTM) with respect to the dilatation factor.
By means of the general eigenvalue problem, we can constitute the principal distortions. At first,
we compute the left Cauchy-Green tensor for the universal transverse Mercator projection modulo
an unknown dilatation parameter according to Corollary
15.5
.
A
1
,A
1
,A
2
, left Cauchy-Green tensor, Universal Transverse Mercator Projection
(UTM) modulo an unknown dilatation parameter).
Corollary 15.5 (
E
The solution of the boundary value problem subject to the integrability conditions of type
Boxes
15.4
and
15.5
constitute the Universal Transverse Mercator Projection (UTM) modulo an
unknown dilatation parameter
ρ
,namely(
15.92
), in the function space of bivariate polynomials.
x
(
l,b
)=
=
ρ
(
x
10
l
+
x
11
lb
+
x
30
l
3
+
x
12
lb
2
+
x
31
l
3
b
+
x
13
lb
3
+
x
50
l
5
+
x
32
l
3
b
2
+
x
14
lb
4
+
+O(6))
,
(15.92)
y
(
l,b
)=
=
ρ
(
y
01
b
+
y
20
l
2
+
y
02
b
2
+
y
21
l
2
b
+
y
03
b
3
+
y
40
l
4
+
y
22
l
2
b
2
+
y
04
b
4
+
y
41
l
4
b
+
y
23
l
2
b
3
+
y
05
b
5
+
+O(6))
.
The coordinates of the left Cauchy-Green deformation tensor
C
l
are represented by
c
11
:=
x
l
+
y
l
,
c
12
:=
c
21
:=
x
l
x
b
+
y
l
y
b
=0
,
(15.93)
c
22
:=
x
b
+
y
h
,
or
x
l
=
ρ
(
x
10
+
x
11
b
+3
x
30
l
2
+
x
12
b
2
+O
lx
(3))
,
y
l
=
ρ
(2
y
20
l
+2
y
21
lb
+O
ly
(3))
,
x
b
=
ρ
(
x
11
l
+2
x
12
lb
+O
bx
(3))
,
(15.94)
y
b
=
ρ
(
y
01
+2
y
02
b
+
y
21
l
2
+3
y
03
b
2
+O
by
(3))
,
and
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