Gauss Surface Normal Coordinates in Geometry and Gravity - Map Projections: Cartographic Information Systems - page 829

⎡ ⎤

+1 + Λ 0 0

⎣

0 1+ Λ

⎦ > 0;

2

(ii) sphere S

r :

0 (J.16)

1 + Λ 0 0

2
A 1 ,A 2

(iii) ellipsoid-of-revolution E :

⎡ ⎤

+1 + A 2 Λ 0 0

⎣

1 + A 2 Λ

⎦ > 0;

0 0 (J.17)

+1 + A 1 Λ 0 0

2
A 1 ,A 2 ,A 2

(iv) triaxial ellipsoid E :

⎡ ⎤

+1 + A 2 A 3 12 Λ 0 0

⎣

+1 + A 1 A 3 Λ

⎦ > 0 .

0 0 (J.18)

+1 + A 1 A 2 Λ 0 0

The solution algorithm presented in Box J.3 determines in the forward step from the first
variational equations (i), (ii), and (iii) the quantities x 1 , x 2 ,and x 3 , and inserts them afterwards
into (iv) as a second forward step. The backward step is organized in first solving for x 4 in a
polynomial equation of type linear, quadratic, or order three. Second, we have to decide whether
our solution fulfills the condition of positivity of the Hesse matrix of second derivatives in order
to discriminate the non-admissible solutions.

Box J.3 (Solution algorithm).

Forward step:
(i) plane :
( a 1 + a 2 + a 3 ) Λ = a 1 X + a 2 Y + a 3 Z + a 4 ; (J.19)

(ii) sphere :
( Λ +1) 2 =( X 2 + Y 2 + Z 2 ) /r 2
⇒ (J.20)

√ X 2 + Y 2 + Z 2 /r,

Λ − =

− 1 −

1+ √ X 2 + Y 2 + Z 2 /r ;
(iii) ellipsoid-of-revolution:

Λ + =

−

(J.21)

A 1 A 2 (1 + A 1 Λ ) 2 (1 + A 2 Λ ) 2 A 2 ( X 2 + Y 2 )(1 + A 1 Λ ) 2 A 1 Z 2 (1 + A 2 Λ ) 2 =0; − −

(iv) triaxial ellipsoid:
A 2 A 3 X 2 (1 + A 1 A 3 Λ ) 2 (1 + A 1 A 2 Λ ) 2 +

(J.22)

+ A 1 A 3 Y 2 (1 + A 2 A 3 Λ ) 2 (1 + A 1 A 2 Λ ) 2 + A 1 A 2 Z 2 (1 + A 2 A 3 Λ ) 2
(1 + A 1 A 3 1 Λ ) 2 −