“Sphere to Cone”: Pseudo-Conic Projections - Map Projections: Cartographic Information Systems - page 444

18-21 Stab-Werner Mapping

In the framework of the Stab-Werner mapping, let us take advantage of the following two postu-
lates.

Postulate.

The North Pole should be mapped to a point.
b =0: r ( Δ =0)=0 . (18.20)

End of Postulate.

Postulate.

An arc on the meridian should be mapped equidistantly.
a = R : r ( Δ )= RΔ. (18.21)

End of Postulate.

Δ−→ 0 = sin Δ

cos Δ
1

α ( Λ, Δ = 0) = lim Λ = lim

Δ−→ 0

Λ = Λ, (18.22)

Δ

− Φ Λ, r ( Δ )= RΔ = R π

Φ α ( Λ, Δ )= sin Δ

Δ

Λ = cos Φ

2 −

(18.23)

π
2

(direct mapping equations) ,

r
R sin R

α, Φ = π

r
R

Λ =

2 −

(18.24)

(inverse mapping equations) .

At this point, we collect the Stab-Werner mapping equations and analyze the principal stretches.
In particular, we observe that the principal stretch components are not directed along the coordi-
nate lines Δ =const ./Φ =const.and Λ = const. because C l is not a diagonal matrix, in general.
We here additionally note that Johannes Werner (1514) in his work “Libellus de quatuor terrarum
orbis in plano figurationibus, Nova translatio primi libri geographiae El. Ptolemai: Neuenberg
(Latin)” remarks: “computed assisted by Johann Stabius”. Furthermore, note that the first pub-
lished map is due to Petrus Aqianus, World Map of Ingolstadt (1530). Moreover, note that the
term cardioform is translated in the form of a heart (Fig. 18.1 ).