“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 226

2 πR 2 = Δ

d x =2 Δ
0

sin x 1+cos 4 2

1+cos 4 2
cos 2

J

sin x
2

J 3 := d x =

cos 2 2

0

=2 Δ
0

1+cos 4 x
2

d x,

tan x
2

x/ 2:= y
⇒

tan x
2

d x =2 tan y (1 + cos 4 y )d y = 1+cos 4 x

2

(5.130)

=2 tan y d y +2 sin y cos 3 y d y =

1

2 cos 4 y

− 2lncos y −

2lncos x

1

2 cos 4 x

= −

2 − 2 ,

ln cos x
2

Δ

2 , cos 4 x

Δ

0 =lncos Δ 0 =cos 4 Δ

2 −

1 ,

2

cos 4 Δ 4lncos Δ

J 3 =1 −

2 − 2 .

(iv) Gnomonic polar azimuthal projection:

sin x 1

d x,

Δ Δ

sin x 1+cos 2 x
cos 4 x

2 πR 2 = 1

J

d x = 1
2

1
cos 2 x

J 4 :=

cos 4 x +

2

0 0

J 4 = 1

6 [1 / cos 3 x ] 0 + 1 2 [1 / cos x ] 0 ,

(5.131)

1
cos 3 Δ −

1 + 1
2

1
cos Δ −

1 ,

J 4 = 1
6

cos 3 Δ
cos 3 Δ

J 4 = 1

1 −

+ 1
2

1 − cos Δ

cos Δ

.

6

(v) Orthographic polar azimuthal projection:

Δ

2 πR 2 = 1

J

sin x (1 + cos 2 x )d x,

J 5 :=

2

0

Δ Δ

J 5 = 1 sin x d x + 1

2

sin x cos 2 x d x,

(5.132)

2

0 0

J 5 = 1

1

6 [cos 3 x ] 0 ,J 5 = 1

cos Δ )+ 1 cos x ] 0 − cos 3 Δ ) .

2 [

−

2 (1

−

6 (1

−