J-1 Projective heights in geometry space: from planar/spherical to ellipsoidal mapping 641
Box J.1 (Projection heights in geometry space: minimal distance mapping).
Geodesics in geometry space:
d
dt
˙
x k
p
˙
x 2 + ˙
y 2 + ˙
z 2
= 0 .
(J.1)
Minimal distance mapping:
S := [X − x(u, v)]
2 + [Y − y(u, v)]
2 + [Z − Z(u, v)]
2 .
(J.2)
Reference surface:
(i)
x(u, v) ∈ P
2
(plane) ,
(ii)
x(u, v) ∈ S
2
r
(sphere) ,
(iii)
x(u, v) ∈ E
2
A 1 ,A 2
(ellipsoid-of-revolution) ,
(iv)
x(u, v) ∈ E
2
A 1 ,A 2 ,A 3
(triaxial ellipsoid) .
(J.3)
Heights in geometry space:
h := X − x(u, v) .
(J.4)
Box J.2 (Projection heights in geometry space: minimal distance mapping, formulae and relations).
Stationary functional:
(i)
L(x
1 , x
2 , x
3 , x
4 ) :=
1
2
X − x
2 + Λ(a 1 x + a 2 y + a 3 z + a 4 ) =
=
1
2
ˆ
(X − x
1 )
2 + (Y − x
2 )
2 + (Z − x
3 )
2
˜
+ x
4 (a 1 x
1 + a 2 x
2 + a 3 x
3 + a 4 ) ,
(J.5)
(ii)
L(x
1 , x
2 , x
3 , x
4 ) := X − x
2 + Λ(x
2 + y
2 + z
2 − r
2 ) =
= (X − x
1 )
2 + (Y − x
2 )
2 + (Z − x
3 )
2 + x
4 [(x
1 )
2 + (x
2 )
2 + (x
3 )
2 − r
2 ] ,
(J.6)
(iii)
L(x
1 , x
2 , x
3 , x
4 ) := X − x
2 + Λ
»
(x
1 )
2 + (x
2 )
2
A
2
1
+
(x
3 )
2
A
2
2
− 1
–
or
L(x
1 , x
2 , x
3 , x
4 ) := (X − x
1 )
2 + (Y − x
2 )
2 + (Z − x
3 )
2 + x
4 `
A
2
2 [(x
1 )
2 + (x
2 )
2 ] + A
2
1 (x
3 )
2 − A
2
1 A
2
2
´
,
(J.7)
(iv)
L(x
1 , x
2 , x
3 , x
4 ) := X − x
2 + Λ
»
(x
1 )
2
A
2
1
+
(x
2 )
2
A
2
2
+
(x
3 )
2
A
2
3
− 1
–
or
L(x
1 , x
2 , x
3 , x
4 ) := X − x
2 + x
4 ˆ
A
2
2 A
2
3 x
2 + A
2
1 A
2
3 y
2 + A
2
1 A
2
2 z
2 − A
2
1 A
2
2 A
2
3
˜
.
(J.8)