“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 186

Postulate.

The polar coordinate α , which is also called azimuth , is identical to the spherical longitude, i.e.
α = Λ .

End of Postulate.

Postulate.

The polar coordinate r depends exclusively on the spherical latitude Φ or on the spherical
colatitude Δ = π/ 2 Φ, i.e. r = x 2 + y 2 = f ( Δ )= f ( π/ 2 − − Φ ). If Φ = π/ 2 or, equivalently,

Δ =0,then f (0) = 0 holds.

End of Postulate.

In last consequence, the general equations of an azimuthal mapping are provided by the following
vector equation:

x
y

= r cos α = f ( Δ )cos Λ .

(5.9)

r sin α f ( Δ )sin Λ

Question: “How do the images of the coordinate line Λ =
constant and the coordinate line Φ = constant look like?”
Answer ( y = x tan Λ and Λ = constant = meridian): “The
image of the meridian Λ = constant under an azimuthal
mapping is the radial straight line.” Answer ( x 2 + y 2 =
r 2 = f 2 ( Δ )and Δ = constant = parallel circle): “The line
of the parallel circle Δ = constant (or Φ = constant) under
an azimuthal mapping is the circle

r of radius r = f ( Δ ).
Such a mapping is called concircular .”

S

Proof ( y = x tan Λ, Λ = constant = meridian).

Solve the first equation towards f ( Δ )= x/ cos Λ and substitute f ( Δ ) in the second equation such
that the following equation holds:

y = f ( Δ )sin Λ = x sin Λ/ cos Λ = x tan Λ. (5.10)

End of Proof ( y = x tan Λ, Λ = constant = meridian).

Proof ( x 2 + y 2 = r 2 = f 2 ( Δ ) ,Δ = constant = parallel circle).

Compute the terms x 2 and y 2 and add the two: