Geography Reference
In-Depth Information
A
1
,A
2
,A
3
with the axes
A
1
>A
2
>A
3
. Fourth, by Box
J.1
, let us here outline a variant
of the analytical treatment of generating geometric heights with respect to (i) the plane
P
ellipsoid
E
2
. (ii)
the sphere
S
r
. (iii) the ellipsoid-of-revolution
E
A
1
,A
2
, and (iv) the triaxial ellipsoid
E
A
1
,A
2
,A
3
by
the minimal distance principle.
By (
J.15
)-(
J.18
), the constraint Lagrangean is defined with
respect to the Euclidean distance
2
/
2 subject to
X
−
x
2
,
x
2
or
r
or
2
A
1
,A
2
2
A
1
,A
2
,A
3
X
, and the follow-
ing constraint. The point
x
is an element of the plane
∈
T
∈
P
S
E
or
E
2
,the
P
r
, the ellipsoid-of-revolution
2
A
1
,A
2
sphere
S
E
, or the triaxial
2
A
1
,A
2
,A
3
ellipsoid
. The constraint enters the Lagrangean
by a Lagrange multiplier
Λ
. The routine of constraint opti-
mization is followed by (
J.9
)-(
J.13
) The focus is on the nor-
mal equations (
J.10
)-(
J.13
), which constitute a system of
algebraic equations of second degree. A solution algorithm
is outlined by (
J.19
)-(
J.23
). In order to guarantee a mini-
mal distance solution, the solution points of the nonlinear
equations of normal type have to be tested with respect to
the second variation, i.e. the positivity of the Hesse matrix
of second derivatives with respect to the unknown coordi-
nates
E
{
x
1
,x
2
,x
3
}
of
p
=
π
(
P
).
First, we alternatively present a second variation of the contruction of projective heights in geom-
etry space by a minimal distance mapping of a topographic point
X
∈
T
2
onto a reference surface
of type (i) plane
P
2
, (ii) sphere
S
r
, (iii) ellipsoid-of-revolution
E
A
1
,A
2
, and (iv) triaxial ellipsoid
E
=ext.
x
(
u, v
) indicates a suitable parameterization
of the surfaces (i)-(iv) by means of coordinates
A
1
,A
2
,A
3
, namely based upon
X
−
x
(
u, v
)
{
u, v
}
. which constitute a chart of the Riemannian
manifold (i)-(iv).
The first variation
δL
(
u, v
)=
δ
X
−
x
(
u,v
)
=0
leads to the normal equations (
J.15
)-(
J.18
), which estab-
lish the orthogonality of type
X
x
(
u, v
)=
h
n
and
∂
x
/∂u
α
(
u, v
)=
t
α
, namely of the normal surfaces
n
and
the surface tangent vector
t
α
for all
α
−
∈{
1, 2
}
.Inpar-
2
by (
J.5
),
ticular, projective heights for (i) the plane
P
2
(ii) the sphere
S
r
by (
J.6
), (iii) the ellipsoid-of-revolution
2
A
1
,A
2
2
A
1
,Λ
2
,A
3
E
by (
J.8
).
With respect to the second variation, besides (
J.15
)-(
J.19
)
as the necessary condition for a minimal distance map-
ping, (
J.19
)-(
J.23
) establishes the suciency condition.
by (
J.7
), and (iv) triaxial ellipsoid
E
The suciency condition has been interpreted by the matrices of the first and second funda-
mental form in
Grafarend and Lohse
(
1991
). In addition, we like to refer to
Bartelme and
Meissl
(
1975
),
Benning
(
1974
),
Froehlich and Hansen
(
1976
),
Heck
(
2002
),
Heikkinen
(
1982
),
Paul
(
1973
),
Penev
(
1978
),
Pick
(
1985
),
Suenkel
(
1976
),
Vincenty
(
1976a
,
1980
), recently J.
Awange and E. Grafarend (
2005
).
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