“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 218

Parameterized inverse mapping:
tan Λ = y

x , cos x 2 + y 2

. (5.97)

R

S 2 R onto the tangent space T N S 2 R : orthographic projection,
Tissot ellipses, polar aspect, graticule 15 ◦ , shorelines

Fig. 5.24. Special perspective map of the sphere

5-247 The Lagrange Projection

The normal general perspective mapping of the sphere reduces to the Polar Stereographic Pro-
jection (UPS) if we specialize H =0or D = R . A special variant already mentioned is achieved
if we choose the South Pole as the perspective center O ∗ (alternatively, the North Pole) and a

2
O

projection plane to coincide with the equatorial frame .InFigs. 5.25 and 5.26 , such a central

perspective mapping is illustrated. In Box 5.17 , the characteristics of such a projection (namely, (i)
the parameterized mapping, (ii) the left principal stretches of the left Cauchy-Green eigenspace,
(iii) the left maximal angular shear, and (iv) the inverse parameterized mapping) are collected.

P

Such a central perspective mapping particularly is associ-
ated with the name of J. L. Lagrange (1736-1813). Note
that his works on map projections are published in A.
Wangerin, Uber Kartenprojectionen, Abhandlungen von
J. L. Lagrange and C. F. Gauss (Verlag W. Engelmann,
Leipzig 1894).

The basic results of the Lagrange projection of the sphere S

2 R to the equatorial plane are collected

in Lemma 5.6 .