Mixed Cylindric Map Projections - Map Projections: Cartographic Information Systems - page 753

Section F-2.

In Sect. F-2 , we start from the setup of the ellipsoidal generalized Lambert projection and the
ellipsoidal generalized Sanson-Flamsteed projection, which are both equiareal. By two lemmas,
we shall present the vertical-horizontal mean of the generalized Lambert projection and the
generalized Sanson-Flamsteed projection of the biaxial ellipsoid.

Section F-3.

The deformation analysis of vertically and horizontally averaged equiareal cylindric mappings is
the topic of Sect. F-3 . We especially compute the left-right principal stretches, their corresponding
eigenvectors/eigenspace, and the maximal left angular distortion.

The following references are appropriate. The equiareal cylindric map projection of the sphere is
addressed to Lambert ( 1772 ), while the equiareal pseudo-cylindrical map projection of the sphere,
namely of sinusoidal type, to N. Sanson (1675), Cossin ( 1570 ), and J. Flamsteed ( 1692 ). Equally
weighted vertical and horizontal components of Lambert and Sanson-Flamsteed map projections
of the sphere have been presented by Foucaut ( 1862 ), Nell ( 1890 ), and Hammer ( 1900 ). The
theory of weighted means of general map projections of the sphere has been critically reviewed
by Tobler ( 1973 ).

F-1 Pseudo-Cylindrical Mapping: Biaxial Ellipsoid onto Plane

2 A,B (ellipsoid-of-revolution, spheroid) with semi-
major axis A and semi-minor axis B based upon local coordinates of type { longitude Λ , latitude
Φ} as surface coordinates summarized in Box F.1 . Second, we set up pseudo-cylindrical mapping
equations of the biaxial ellipsoid E

First, we refer to a chart of the biaxial ellipsoid E

2 A,B onto the Euclidean plane { R 2 ,δ μν } in terms of surface

normal ellipsoidal { longitude Λ , latitude Φ} , in particular

cos Φ

x = x ( Λ, Φ )= AΛ 1 g ( Φ ) ,

(F.1)

E 2 sin 2 Φ

−

y = y ( Φ )= f ( Φ ) ,

2

x := { x ∈ R | ax + by + c =0 } . (F.2)

The structure of the pseudo-cylindrical mapping equations with unknown functions f ( Φ )and
g ( Φ ) is motivated by the postulate that for g ( Φ ) = 1 parallel circles of

2

E

A,B should be mapped

2 ,δ μν }

equidistantly onto .([ δ μν ] denotes the unit matrix, all indices run from one to two.) Note

that for zero relative eccentricity E = 0, we arrive at the pseudo-cylindrical mapping equations
of the sphere

{ R

2

S

R .