“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 198

a bundle of projection lines, in particular, half straights. Second, we place the projection plane
(i) at maximum distance from

O ∗ at the North Pole to coincide with the tangential plane T N S 2 R ,

(ii) at the center

2 R as the equatorial plane, and (iii) at minimum distance from
O ∗ at the South Pole to coincide with the tangential plane T N S

O of the sphere S

2 R . Note that the projection lines

2

intersect the sphere

R at P , while the projection plane is intersected at p . The perspective center
O ∗ is at distance D from the origin

S

2

O ∗ ,

O of the sphere S

R or at height H above S, measured by S

such that D = R + H holds.

Question: “How to find the polar coordinate r = f ( Δ )in
Figs. 5.10 , 5.11 ,and 5.12 ,where Δ = π/ 2 Φ is the spher-

ical colatitude and Φ is the spherical latitude of the point
P

−

2

R ?” Answer: “Consult the sub-sections that follow,
which compactly present the case studies for the individual
geometrical situations.”

∈ S

O ∗ N / O ∗ Q is the fundament for the

Note that in all these cases the perspective ratio r/QP =

answer to the well-posed question.

Fig. 5.10. Spherical vertical section, general normal perspective mapping of the sphere to the tangential plane
at the North Pole, projection plane at maximal distance

5-241 Case 1: Northern Tangential Plane (Tangential Plane at Maximal Distance)

This situation is shown in Fig. 5.10 . With reference to Boxes 5.7 and 5.8 ,wederivethegeneral
form of the parameterized mapping r = f ( Δ ). Note that Q = π ( P ) is the point generated by an