From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 104

Box 1.56 (The canonical representation of left angular shear and right angular shear. Special
case: G 12 =0, c 12 =0and g 12 =0, C 12 =0).

Left and right angular parameters Ψ, and Ψ r :
cos Ψ l = C 11 d u
d s

versus √ c 11 d U
d S

d s
d S

d S
d s =cos Ψ r ,

cos Ψ l = √ G 11 d u

1
λ

√ c 11 d U 1

Λ =cos Ψ r ,

versus

d s d S

(1.331)

sin Ψ l = C 22 d v
d s

versus √ c 22 d V
d S

d s
d S

d S
d s =sin Ψ r ,

sin Ψ l = C 22 d v

1
Λ =sin Ψ r .
Left and right stretches, left and right principal stretches:

1
λ

versus √ c 22 d V

d s d S

λ 2 ;= d S 2
d s 2

versus Λ 2 := d s 2

d S 2 ,

λ 1 = C 11 versus Λ 1 = c 11

G 11 .

(1.332)

g 11

λ 2 = C 22 versus Λ 2 = c 22

G 22 , g 22

G 11 U =cos Ψ 1

cos Ψ r = √ g 11 u

⇒ ⇒

C 11
g 11

cos Ψ l c 11
G 11

1
λ

1
Λ =cos Ψ r ,

cos Ψ l =cos Ψ r versus (1.333)

C 22
g 22

sin Ψ l c 22
G 22

1
λ

1
Λ =sin Ψ r ,

sin Ψ l =sin Ψ r

versus

cos Ψ l =cos Ψ r λ 1
λ

cos Ψ l Λ 1

versus

Λ =cos Ψ r ,

sin Ψ l =sin Ψ r λ 2 sin Ψ l Λ 2 versus

Λ =sin Ψ r ,

(1.334)

λ

tan Ψ l = λ 2

Λ 2
Λ 1 tan Ψ l =tan Ψ r . λ 1 tan Ψ r

versus

Left and right angular shear, left and right angulardistortion:
tan( Ψ l − Ψ r )=tan l versus tan r =tan( Ψ r − Ψ l ) ,

tan l = versus tan r =

tan Ψ r
1+tan Ψ l tan Ψ r

tan Ψ l − tan Ψ l

1+tan Ψ r tan Ψ l ,

tan Ψ r −

tan l = tan Ψ l versus tan r = tan Ψ r

Λ 2 Λ − 1 tan Ψ l
1+ Λ 2 Λ − 1 tan 2 Ψ l

λ 2 λ − 1 tan Ψ r
1+ λ 2 λ − 1 tan 2 Ψ r

− −

, (1.335)