From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 87

Fig. 1.29. Bonne-pseudo-conic projection, with shorelines of a spherical Earth, equidistant mapping of the
line-of-contact of a circular cone, “quasicordiform”. Tissot ellipses of distortion. (According to Rigobert Werner)

We have already defined areomorphism, namely areal distortion, in order to present here an
equivalence theorem that relates areomorphism to a special partial differential equation whose
solution guarantees an equiareal mapping. In particular, we make a “canonical statement” about
the product of left and right principal stretches to be one. Furthermore, we specify the equiareal
mapping for a right manifold

2 2 ,δ μν }

{ M

r ,g μν }

= { R to be Euclidean.

Theorem 1.14 (Areomorphism M

l

→ M

r , equiareal mapping).

Let f :

r be an orientation preserving equiareal mapping. Then the following conditions
(i)-(iv) are equivalent.

M

l

→ M

(i)

det[G l ]d U

V = det[G r ]d u ∧ ∧ d v.

det[C r ]=det[G r ] , det[C l ]=det[G l ] , det[G r − 2E r ]=det[G r ] , det[G l +2 E l ]

=det[G l ] . (1.272)

(iii)
Λ 1 Λ 2 =1 , λ 1 λ 2 =1 .
(iv)