638 J Gauss surface normal coordinates in geometry and gravity space
J-1 Projective heights in geometry space: from planar/spherical to
ellipsoidal mapping
First, we outline how points in {R
3 , δ kl } are connected by geodesics, namely by straight lines which are
derived from a variational principle. The general solution of the differential equations of a geodesic,
in particular, in terms of an affine parameter of its length, is represented by a linear one-dimensional
manifold embedded into {R
3 , δ kl }. Second, based upon geodesics in {R
3 , δ kl }, in Fig. J.1, we introduce
the orthogonal projection of a point P (the peak of a mountain, the top of a tower) onto a horizontal
plane through P 0 by p = π(P ) generating the height pP of P with respect to the plane P
2
0 through P 0 .
Alternatively, we may interpret the orthogonal projection p = π(P ) along a geodesic/straight line as a
minimal distance mapping of P with respect to P 0 generating p = π(P ). Third, by Fig. J.2, we illustrate
the minimal distance mapping of a topographic point P ∈ T
2 as an element of the topographic surface
(two-dimensional Riemann manifold) of the Earth along a geodesic/straight line onto a plane P
2
which may be chosen as the horizontal plane at some reference point. In this way, we generate the
orthogonal projection p = π(P ) and the length of the shortest distance pP , called the geometric
height of P with respect to P
2 . The choice of the height pP is very popular in photogrammetric and
engineering surveying. By contrast, by Figs. J.3 and J.4, we illustrate the minimal distance mapping
of a point P onto p ∈ S
2
r or p ∈ E
2
A 1 ,A 2
or p ∈ E
2
A 1 ,A 2 ,A 3
along a geodesic/straight line through P .
Let us here assume that the reference surface is no longer the plane P
2 , but the sphere S
2
r of radius
r or the ellipsoid-of-revolution E
2
A 1 ,A 2
of semi-major axis A 1 and semi-minor axis A 2 or the triaxial
ellipsoid E
2
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
2 , (ii) the sphere
S
2
r , (iii) the ellipsoid-of-revolution E
2
A 1 ,A 2
, and (iv) the triaxial ellipsoid E
2
A 1 ,A 2 ,A 3
by the minimal
distance principle.
Important!
By (J.15)–(J.18), the constraint Lagrangean is defined with respect to the Euclidean distance
X − x
2 /2 subject to X ∈ T
2 , x ∈ P
2 or S
2
r or E
2
A 1 ,A 2
or E
2
A 1 ,A 2 ,A 3
, and the following constraint. The point x is an element of the plane P
2 , the sphere S
2
r , the ellipsoid-of-revolution
E
2
A 1 ,A 2
, or the triaxial ellipsoid E
2
A 1 ,A 2 ,A 3
. The constraint enters the Lagrangean by a Lagrange multiplier Λ. The routine of constraint optimization is followed by (J.9)–(J.13) The
focus is on the normal 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 minimal 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 coordinates {x
1 , x
2 , x
3
} of p = π(P ).
First, we alternatively present a second variation of the contruction of projective heights in geometry
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
2
r , (iii) ellipsoid-of-revolution E
2
A 1 ,A 2
, and (iv) triaxial ellipsoid E
2
A 1 ,A 2 ,A 3
,
namely based upon X − x(u, v) = ext. x(u, v) indicates a suitable parameterization of the surfaces
(i)–(iv) by means of coordinates {u, v}, which constitute a chart of the Riemannian manifold (i)–(iv).
Important!
The first variation δL(u, v) = δX − x(u, v) = 0 leads to the normal equations (J.15)–
(J.18), which establish the orthogonality of type X − x(˜ u, ˜
v) = hn 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}. In
particular, projective heights for (i) the plane P
2 by (J.5), (ii) the sphere S
2
r by (J.6), (iii)
the ellipsoid-of-revolution E
2
A 1 ,A 2
by (J.7), and (iv) triaxial ellipsoid E
2
A 1 ,A 2 ,A 3
by (J.8).
With respect to the second variation, besides (J.15)–(J.19) as the necessary condition for a
minimal distance mapping, (J.19)–(J.23) establishes the sufficiency condition.
The sufficiency condition has been interpreted by the matrices of the first and second fundamental
form in E. Grafarend and P. Lohse (1991). In addition, we like to refer to N. Bartelme and P. Meissl
(1975), W. Benning (1974), H. Fr¨ ohlich and H. H. Hansen (1976), B. Heck (2002), M. Heikkinen (1982),
M. K. Paul (1973), P. O. Penev (1978), M. Pick (1985), H. S¨ unkel (1976), T. Vincenty (1976, 1980),
recently J. Awange and E. Grafarend (2005).
J-1 Projective heights in geometry space: from planar/spherical to
ellipsoidal mapping
First, we outline how points in {R
3 , δ kl } are connected by geodesics, namely by straight lines which are
derived from a variational principle. The general solution of the differential equations of a geodesic,
in particular, in terms of an affine parameter of its length, is represented by a linear one-dimensional
manifold embedded into {R
3 , δ kl }. Second, based upon geodesics in {R
3 , δ kl }, in Fig. J.1, we introduce
the orthogonal projection of a point P (the peak of a mountain, the top of a tower) onto a horizontal
plane through P 0 by p = π(P ) generating the height pP of P with respect to the plane P
2
0 through P 0 .
Alternatively, we may interpret the orthogonal projection p = π(P ) along a geodesic/straight line as a
minimal distance mapping of P with respect to P 0 generating p = π(P ). Third, by Fig. J.2, we illustrate
the minimal distance mapping of a topographic point P ∈ T
2 as an element of the topographic surface
(two-dimensional Riemann manifold) of the Earth along a geodesic/straight line onto a plane P
2
which may be chosen as the horizontal plane at some reference point. In this way, we generate the
orthogonal projection p = π(P ) and the length of the shortest distance pP , called the geometric
height of P with respect to P
2 . The choice of the height pP is very popular in photogrammetric and
engineering surveying. By contrast, by Figs. J.3 and J.4, we illustrate the minimal distance mapping
of a point P onto p ∈ S
2
r or p ∈ E
2
A 1 ,A 2
or p ∈ E
2
A 1 ,A 2 ,A 3
along a geodesic/straight line through P .
Let us here assume that the reference surface is no longer the plane P
2 , but the sphere S
2
r of radius
r or the ellipsoid-of-revolution E
2
A 1 ,A 2
of semi-major axis A 1 and semi-minor axis A 2 or the triaxial
ellipsoid E
2
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
2 , (ii) the sphere
S
2
r , (iii) the ellipsoid-of-revolution E
2
A 1 ,A 2
, and (iv) the triaxial ellipsoid E
2
A 1 ,A 2 ,A 3
by the minimal
distance principle.
Important!
By (J.15)–(J.18), the constraint Lagrangean is defined with respect to the Euclidean distance
X − x
2 /2 subject to X ∈ T
2 , x ∈ P
2 or S
2
r or E
2
A 1 ,A 2
or E
2
A 1 ,A 2 ,A 3
, and the following constraint. The point x is an element of the plane P
2 , the sphere S
2
r , the ellipsoid-of-revolution
E
2
A 1 ,A 2
, or the triaxial ellipsoid E
2
A 1 ,A 2 ,A 3
. The constraint enters the Lagrangean by a Lagrange multiplier Λ. The routine of constraint optimization is followed by (J.9)–(J.13) The
focus is on the normal 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 minimal 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 coordinates {x
1 , x
2 , x
3
} of p = π(P ).
First, we alternatively present a second variation of the contruction of projective heights in geometry
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
2
r , (iii) ellipsoid-of-revolution E
2
A 1 ,A 2
, and (iv) triaxial ellipsoid E
2
A 1 ,A 2 ,A 3
,
namely based upon X − x(u, v) = ext. x(u, v) indicates a suitable parameterization of the surfaces
(i)–(iv) by means of coordinates {u, v}, which constitute a chart of the Riemannian manifold (i)–(iv).
Important!
The first variation δL(u, v) = δX − x(u, v) = 0 leads to the normal equations (J.15)–
(J.18), which establish the orthogonality of type X − x(˜ u, ˜
v) = hn 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}. In
particular, projective heights for (i) the plane P
2 by (J.5), (ii) the sphere S
2
r by (J.6), (iii)
the ellipsoid-of-revolution E
2
A 1 ,A 2
by (J.7), and (iv) triaxial ellipsoid E
2
A 1 ,A 2 ,A 3
by (J.8).
With respect to the second variation, besides (J.15)–(J.19) as the necessary condition for a
minimal distance mapping, (J.19)–(J.23) establishes the sufficiency condition.
The sufficiency condition has been interpreted by the matrices of the first and second fundamental
form in E. Grafarend and P. Lohse (1991). In addition, we like to refer to N. Bartelme and P. Meissl
(1975), W. Benning (1974), H. Fr¨ ohlich and H. H. Hansen (1976), B. Heck (2002), M. Heikkinen (1982),
M. K. Paul (1973), P. O. Penev (1978), M. Pick (1985), H. S¨ unkel (1976), T. Vincenty (1976, 1980),
recently J. Awange and E. Grafarend (2005).