From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 86

Box 1.39 (Areal distortion, representations of its multiplicative and additive deformation
measures).

(i)

2
l

2

M → M

r :

det[G l ]d U

d V = det[G r ]d u

∧ ∧ d v,

det[G r ]d u

d v = det[ G l ]d U

∧ ∧ d V ; (1.269)

det[G l ]d U

d V = det[G r

∧ − 2E r ]d u ∧ d v,

det[G r ]d u

d v = det[G l

∧ − 2E l ]d U ∧ d V ;

det[G l ]d U

1
det[F r ] λ 1 λ 2 d u

∧ d V = ∧ d v,

det[G r ]d u

1
det[F l ] Λ 1 Λ 2 d U

∧ d v = ∧ d V ;

det[G l ]d U

2 2 ,δ μν

{ M

r ,g μν

} = { R }⇒ ∧ d V = λ 1 λ 2 d u ∧ d v.

(ii)

Φ l = det[C l G − l ]= Λ 1 Λ 2 ,

(1.270)

Φ r = det[C r G − r ]= λ 1 λ 2 .

(iii)

S l r = det[C l ]

G l d U

d V,
S l r = det[G r ] − √ G r d u ∧ d v ;

− ∧

1
det[F l ] d U

S l r =( Λ 1 Λ 2 − 1) ∧ d V, (1.271)

1
det[F r ] d u

S r l =( λ 1 λ 2 − 1) ∧ d v ;

2 2 ,δ μv }⇒

{ M

r ,g μν }

= { R S r l =( λ 1 λ 2 − 1)d u ∧ d v.

To give you again some breathing time, please enjoy Fig. 1.29 , which presents the “quasicordi-
form” Bonne-pseudo-conic projection.

1-12 Equivalence Theorem of Equiareal Mapping

The equivalence theorem of equiareal mapping from the left to the right two-dimensional
Riemann manifold (areomorphism).