“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 199

Fig. 5.11. Spherical vertical section, the general normal perspective mapping of the sphere to the equatorial
plane. The geometrical details

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orthogonal projection of the point P ∈ S

R onto the axis of symmetry North-Pole-South-Pole. Let

us refer to the following identities.

Identity (i):
QP = R cos Φ.
Identity (ii):
O ∗ N= R + D =2 R + H. (5.45)

Identity (iii):
O ∗ Q =

O ∗ O

Q =
= D + R sin Φ = R (1 + sin Φ )+ H.

+ O

Solving the perspective ratio for r , we are finally led to r = f ( Δ ). Such a representation of the
radial function f ( Δ ) is supplemented by the computation of f ( Δ ), a formula needed for the
analysis of the left principal stretches.

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Box 5.7 (Basics of the perspective ratio, northern tangential plane T N S

R ).

Basic ratio:
r

QP = O ∗ N O ∗ Q .

(5.46)