Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 253

{ C l , G l } is canonically characterized by the left principal stretches Λ 1 and Λ 2 in their general

form.

Box 8.2 (“Ellipsoid-of-revolution to plane”, distortion analysis, azimuthal projection, left
principal stretches).

Parameterized mapping:
α = Λ,
r = f ( Δ ) , (8.14)

x = r cos α = f ( Δ )cos Λ,
y = r sin α = f ( Δ )sin Λ.

Left Jacobi matrix:
J l := D Λ xD Δ x

= − .

f ( Δ )sin Λf ( Δ )cos Λ
+ f ( Δ )cos Λf ( Δ )sin Λ

(8.15)

D 4 yD Δ y

Left Cauchy-Green matrix(G r =I 2 ):
C l =J l G r J l = f 2 ( Δ )0

.

(8.16)

0 f 2 ( Δ )

Left principal stretches:
Λ 1 =+ c 11

G 11 = f ( Δ ) √ 1

E 2 cos 2 Δ
A 1 sin Δ

−

,

(8.17)

Λ 2 =+ c 22

G 22 = f ( Δ )(1

E 2 cos 2 Δ ) 3 / 2
A 1 (1

−

.

− E 2 )

Left eigenvectors of the matrix pair { C l , G l } :

D Λ X

C 1 = E Λ =

D A X

(Easting) , (8.18)

D Φ X

C 2 = E Φ =

D Φ X

(Northing).

Next, we specialize the general azimuthal mapping to generate an equidistant mapping, a series
of conformal mappings (stereographic projections), and an equiareal mapping.