Geodesics - Map Projections: Cartographic Information Systems - page 735

32 R 4

(4 R 2 +( q 1 ) 2 +( q 2 ) 2 ) 3 q μ

p μ = − ∀ μ =1 , 2 .

(ii) Universal Mercator Projection (UM):
q 1 =0 , q 1 + 1
R

sinh( q 2 /R )

cosh 3 ( q 2 /R ) =0 ,
q 1 = p 1 , q 2 = p 2 , (E.65)

p 1 =0( p 1 cyclic) ,p 1 =const .,

sinh( q 2 /R )
cosh 3 ( q 2 /R ) .

1
R

p 2 = −

(iii) Universal Transverse Mercator Projection (UTM):

q 1 =0 ,

1 − tanh 2 q 2

2 R sin 2arcsin tanh q 2

1

q 2 +

R =0 , R

q 1 = p 1 , q 2 = p 2 ,

(E.66)

p 1 =0( p 1 cyclic) ,p 1 =const .,

1

2 R sin 2arcsin tanh q 2

1

tanh 2 q 2 p 2 = − −

R .

R

(iv) Universal Conic Projection (UC) (variant one and variant two):

c ( q 1 ) 2 +( q 2 ) 2 1 /n 2 c 1 /n n 3 sin 2 arcsin 1

1

q μ − ×

c ( q 1 ) 2 +( q 2 ) 2 2 /n − 1

1+ 1

[( q 1 ) 2 +( q 2 ) 2 ] (1 −n ) / 2 n q μ =0 ,

×

q 1 = p 1 , q 2 = p 2 , (E.67)

c ( q 1 ) 2 +( q 2 ) 2 1 /n

2 c 1 /n n 3 sin 2 arctan 1

1

p μ = ×

c ( q 1 ) 2 +( q 2 ) 2 2 /n − 1

× 1+ 1

[( q 1 ) 2 +( q 2 ) 2 ] (1 −n ) / 2 n q μ .