Geodetic Mapping - Map Projections: Cartographic Information Systems - page 475

Table 20.1 Various definitions of (Riemann) polar and normal coordinates.

orthonormal two-leg (Riemann) polar/normal

(Cartan two-leg) azimuth coordinates

{ East, North } :

East azimuth x = r cos α, y = r sin α

X L
X L

X B
X B

= C 1 , = C 2

North azimuth

x ∗ = r cos α ∗ = r sin α,

{ North, East } :

(left oriented),

X / B
X B

= C 1 ,

X L
X L

= C 2 y ∗ = r sin α ∗ = r cos α

α ∗ =90 ◦ −

α
South azimuth

x ∗∗ = r cos α ∗∗ =

{ South, East } : − r sin α,

(right oriented),

X B
X B

= C ∗∗
1

X L
X L

= C ∗∗
2

y ∗∗ = r sin α ∗∗ = r cos α − ,

α ∗∗ =90 ◦ + α

Fig. 20.3. Oblique tangential plane T U 0 E 2 A 1 ,A 2 ,Cartanframe C 1 (East) and C 2 (North) at point P 0 ( U 0 )

Let us discuss how to relate the polar or normal tangential coordinates { α,r } to those coor-

2
A 1 ,A 2

dinates which parameterize E ,here { longitude L , latitude B } or { meta-longitude U ,meta-

2
A 1 ,A 2

latitude V } , respectively. At first, let us identify the curve C :[0 , ∞ ] →{ E ,G KL } with a

geodesic defined by { κ g =0,( 20.42 ) } . Preparatory is Corollary 20.3 .