“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 221

2 =tan π

= 1

R x 2 + y 2 ,

tan Δ

Φ
2 4 −

2tan 2
1+tan 2 2

2sin Δ

2 cos Δ 2 =

=sin Δ =cos Φ,

x 2 + y 2
1+ x 2 + y 2
R 2

x 2 + y 2
R 2 + x 2 + y 2 ,

2tan 2
1+tan 2 2

= 2
R

cos Φ = =2 R (5.110)

tan 2 2
1+tan 2 2

tan 2 2
1+tan 2 2

cos 2 Δ

2 − sin 2 Δ

1
1+tan 2 2 −

= 1

−

2 =

=cos Δ =sin Φ,

x 2 + y 2
R 2
1+ x 2 + y 2

sin Φ = 1 − tan 2 2
1+tan 2 2

= 1 − = R 2

( x 2 + y 2 )
R 2 +( x 2 + y 2 ) ,

−

R 2

x 2 + y 2
R 2 + x 2 + y 2 ,

cos Φ =2 R

(5.111)

sin Φ = R 2

( x 2 + y 2 )
R 2 +( x 2 + y 2 ) .

−

2
R

3 :

S ⊂ E

⎡ ⎤

X
Y
Z

⎣ ⎦ ,

X ( Λ, Φ, R )=[ E 1 , E 2 , E 3 ] (5.112)

x
R 2 +( x 2 + y 2 ) ,

X = R cos Φ cos Λ =2 R

y
R 2 +( x 2 + y 2 ) ,

Y = R cos Φ sin Λ =2 R (5.113)

Z = R sin Φ = R R 2

− ( x 2 + y 2 )
R 2 +( x 2 + y 2 ) .

5-25 What Are the Best Polar Azimuthal Projections
of “Sphere to Plane”?

Most textbooks on map projections list those many azimuthal projections of the “sphere to plane”
without taking any decision of which one may be the best. Indeed, for such a decision, we need
an objective criterion, and we choose it according to Chaps. 1 and 2 , i.e. we choose the distortion
energy over a spherical cap being covered by the chosen azimuthal projection of the sphere S

2 R to

2 R or the plane 2

O

the tangent space T N . In order to prepare us for a rational decision of the best

polar azimuthal projection “sphere to plane”, in Table 5.2 , we have tabulated a variety of values for
the left principal stretches Λ 1 ( Δ ) along the parallel circle and Λ 2 ( Λ ) along the meridian for the area

S P