From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 103

Box 1.54 (Angular parameters Ψ l and Ψ r ).

“Left” :
X := ∂ ∂U d d S + ∂ ∂V d d S ;
cos Ψ l = X C 1 c os Ψ l = X | C 1 ,

“Right” :
x := ∂ ∂u d d s + ∂ ∂v d d s ;

x x |

cos Ψ r = c 1 co s Ψ r = c 1 ,

cos Ψ l = √ G 11 U ;

cos Ψ r = √ g 11 u ;

(1.326)

sin Ψ l =cos 2 − Ψ l = sin Ψ r =cos 2 − Ψ r =

= X C 2 cos 2 − Ψ l = X | C 2 ,

= x c 2 cos 2 − Ψ r = x | c 2 ,

sin Ψ l = √ G 22 V .

sin Ψ r = √ g 22 v .

Box 1.55 (Transformation of angular parameters Ψ l and Ψ r . Special case: G 12 =0, c 12 =0,
u ( U ), v ( V )versus U ( u ), V ( v )).
cos Ψ l = √ G 11 U = √ G 11 U 2 ,

cos Ψ r = √ g 11 u = g 11 u 2 ,

sin Ψ r = √ g 22 v = g 22 v 2 ,
u = d d s ,v = d d s .

sin Ψ l = √ G 22 V = √ G 22 V 2 .

(1.327)

U = d d u d d s d s

d S ,V = d d v d d s d s d S u = d u

d U
d S

d S

d s ,v = d v

d V
d S

d S
d s d U d V

⇒ ⇒

cos Ψ l = G 11 d d u 2 u d s

d S , cos Ψ l = g 11 d u

d U 2 U d d s ,

(1.328)

sin Ψ l = G 22 d d v 2 v d s sin Ψ r = g 22 d v

d V 2 V d d s d S

cos Ψ l = C 11 u d s

d S , sin Ψ l = C 22 v d s d S .

cos Ψ r = √ c 11 U d S

d s , sin Ψ r = √ c 22 V d S d s .

Corollary 1.21 (The canonical representation of left angular shear and right angular shear. Spe-
cial case: G 12 =0, c 12 =0and g 12 =0, C 12 =0).
Let l := Ψ l − Ψ r and r := Ψ r − Ψ l , respectively, denote left and right angular shear, a measure

of the deviation of the mapping M

l

→ M

r from conformality. Then a canonical representation of

the angular parameters Ψ l and Ψ r as well as of the angular shear parameters l and r is

tan Ψ l = λ 2

λ 1 tan Ψ r versus tan Ψ r = Λ 1

Λ 2 tan Ψ l ,

(1.329)

tan l =( Λ 1 − Λ 2 ) versus tan r =( λ 1 − λ 2 )

tan Ψ l
Λ 1 + Λ 2 tan 2 Ψ l

tan Ψ r

λ 1 + λ 2 tan 2 Ψ r .

(1.330)

End of Corollary.