J-1 Projective heights in geometry space: from planar/spherical to ellipsoidal mapping 639
P 0
X
x
P
p = π(P )
X − x = h
P 0 p =
q
P 0 p
2 − pP
2 = P 0 P
q
1 −
`
pP /P 0 P
´ 2 = P 0 P
`
1 − pP /2P 0 P
´
Fig. J.1. Projective heights in geometry space, orthogonal projection p = π(P ) of a topographic point X ∈ T
2
onto a local horizontal plane X 0 ∈ T
2 , minimal distance mapping δX − x
2 = 0.
Indeed, for a suitable choice of surface parameters/surface coordinates of a chart, the unconstrained
optimization problem may be preferable to constraint optimization, however, which we do not want
to treat here.
X
x
Fig. J.2. Projective heights in geometry space, minimal distance mapping with respect to a reference plane
P
2 at X 0 ∈ T
2 , orthogonal projection p = π(P ).
P 0
X
x
P
p = π(P )
X − x = h
P 0 p =
q
P 0 p
2 − pP
2 = P 0 P
q
1 −
`
pP /P 0 P
´ 2 = P 0 P
`
1 − pP /2P 0 P
´
Fig. J.1. Projective heights in geometry space, orthogonal projection p = π(P ) of a topographic point X ∈ T
2
onto a local horizontal plane X 0 ∈ T
2 , minimal distance mapping δX − x
2 = 0.
Indeed, for a suitable choice of surface parameters/surface coordinates of a chart, the unconstrained
optimization problem may be preferable to constraint optimization, however, which we do not want
to treat here.
X
x
Fig. J.2. Projective heights in geometry space, minimal distance mapping with respect to a reference plane
P
2 at X 0 ∈ T
2 , orthogonal projection p = π(P ).