Mercator Projection and Polycylindric Projection - Map Projections: Cartographic Information Systems - page 813

Λ 1 = Λ 2 = Λ S = ρ 2 (1 + μ ) ∀ μ 1 ,

(I.24)

Λ 1 = Λ 2 = Λ S = ρ (1 + μ

2 )+O( μ 2 ) ∀μ 1 .

J l A :

1
2 S

1) 2 +( Λ 2 1) 2 ]d S

J l A := [( Λ 1 − − ∀ Λ 1 = Λ 2 = Λ S ,

S

(I.25)

(1 + μ ) ρ 2

2 1+ μ
2

ρ d S +O( μ 2 ) ,

J l A := 1
S

1) 2 d S =1+ 1
S

( Λ S − −

S S

J l A ( ρ )=0
⇔

ρ =
S

d S 1+ μ

2

(1 + μ )d S +O( μ 2 )

S

⇔ (I.26)

ρ = 1+ 1
2 S

μ d S 1+ 1
S

μ d S +O( μ 2 )

S S

1
2 S

μ d S +O( μ 2 ) . =1 −

S

J l :

J l := 1
S

( Λ 1 Λ 2 − 1)d S ∀ Λ 1 = Λ 2 = Λ S ,

S

(I.27)

J l := 1
S

( Λ S −

1)d S,

S

J l =0
⇔

1
S

[ ρ 2 (1 + μ )

− 1]d S =0

S

⇔

1

ρ 2 ( J l =0)=

1+ S S μ d S

(I.28)

1
S

μ d S +O( μ 2 )

=1 −

S

⇔

1
2 S

μ d S +O( μ 2 ) .

ρ ( J l =0)=1 −

S

ρ ( J l A =min)= ρ ( J l =0) .

End of Proof.