330 15 “Ellipsoid-of-revolution to cylinder”: transverse aspect
15-4 Principal distortions and various optimal designs (UTM mappings)
Principal distortions and various optimal designs of the Universal Transverse Mercator Projection (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.
Corollary 15.5 (E
2
A 1 ,A 1 ,A 2
, left Cauchy–Green tensor, Universal Transverse Mercator Projection (UTM)
modulo an unknown dilatation parameter).
The solution of the boundary value problem subject to the integrability conditions of type Box 15.4
and Box 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)
,
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)
.
(15.92)
The coordinates of the left Cauchy–Green deformation tensor C l are represented by
c 11 := x
2
l + y
2
l ,
c 12 := c 21 := x l x b + y l y b = 0 ,
c 22 := x
2
b + y
2
b ,
(15.93)
or
x l = ρ
x 10 + x 11 b + 3x 30 l
2 + x 12 b
2 + O lx (3)
,
y l = ρ
2y 20 l + 2y 21 lb + O ly (3)
,
x b = ρ
x 11 l + 2x 12 lb + O bx (3)
,
y b = ρ
y 01 + 2y 02 b + y 21 l
2 + 3y 03 b
2 + O by (3)
,
(15.94)
and
c 11 = ρ
2
x
2
10 + (4y 20 + 6x 10 x 30 )l
2 + 2x 10 x 11 b + x
2
11 b
2 + O l (3)
,
c 22 = ρ
2
y
2
01 + (x 11 + 2y 01 y 21 )l
2 + 4y 01 y 02 b + (4y
2
02 + 6y 01 y 03 )b
2 + O b (3)
.
(15.95)
End of Corollary.