Geography Reference
In-Depth Information
H-22 The Explicit Form of the Mapping Equations Generating
an Equiareal Map
We here consider the explicit form of the mapping equations generating an equiareal map from a
left biaxial ellipsoid to a right biaxial ellipsoid with respect to a transverse frame of reference and
a change of scale. Let us begin with the setup ( H.47 )ofBox H.2 of general mapping equations
x = c 1 x ( Λ )and y = c 2 y ( Λ ) in a transverse frame of reference and under a change of
scale with respect to gauge constants
Λ }
which characterize a
point on the right ellipsoid-of-revolution are related via ( H.48 ) ( Λ ) ( Φ ) } to the coordinates
{Λ, Φ} characteristic for a point on the left ellipsoid-of-revolution. Note that right ellipsoidal
longitude/latitude depend only on left ellipsoidal longitude/latitude. In addition, we assume a
coincidence between the semi-major axis A 1 and A 1 , respectively, the semi-minor axis A 2 and
A 2 between right and left E
{
c 1 ,c 2
}
. Those coordinates
{
A 1 ,A 2 , respectively, expressed by ( H.49 ). Next, by ( H.50 )
and ( H.51 ), we subscribe the differential relations d Λ / d Λ = c 3 subject to Λ ( Λ =0)=0and
d Φ / d Φ = c 4 (1
A 1 ,A 2 and E
E 2 sin 2 Φ ) 2 cos Φ/ [(1
E 2 sin 2 Φ ) 2 cos Φ ] subject to Φ ( Φ = 0) = 0 with respect
to gauge constants
.( H.45 ) has motivated ( H.51 ). The detailed computation of the left
Cauchy-Green tensor via ( H.52 ), ( H.53 ), ( H.54 ), and ( H.55 ) leads us to the postulate ( H.56 )of
an equiareal mapping. Indeed, we take advantage via ( H.56 ), ( H.57 ), and ( H.58 )ofthefactthat
{
{
c 3 ,c 4 }
x ( Λ ) ,y ( Λ )
}
is already an equiareal mapping. Thus, we may consider the transfor-
x ,y }→{
mation
as a change from one equiareal chart to another equiareal chart (a:a:
cha-cha-cha). ( H.59 ) is a representation of the postulate of an areomorphism which leads by
subscribing d Λ / d Λ = c 3 to the explicit form of d Φ / d Φ of type ( H.52 ), too. Indeed, we do not
have to postulate ( H.51 )! In order to guarantee an equiareal mapping
{
x, y
}
{
Λ, Φ
}→{
x, y
}
, the gauge
constants have to fulfill ( H.60 ), namely c 1 c 2 c 3 c 4 =1.
Box H.2 (The equiareal mapping from a left biaxial ellipsoid to a right biaxial ellipsoid with
respect to a transverse frame of reference and a change of scale (the Hammer projection of
the ellipsoid-of-revolution)).
x = c 1 x ( Λ ) ,y = c 2 y ( Λ ) ,
(H.47)
Λ ( Λ ) ( Φ ) ,
(H.48)
subject to
A 1 = A 1 , A 2 = A 2 , E = E,
(H.49)
Λ A = d Λ
d Λ = c 3 ,
(H.50)
Φ Φ = d Φ
E 2 sin 2 Φ ) 2
d Φ = c 4 (1
cos Φ
cos Φ .
(H.51)
E 2 sin 2 Φ ) 2
(1
The left Cauchy-Green deformation tensor:
∂Λ = c 1 ∂x
d Λ
∂Φ = c 1 ∂x
d Φ
x Λ = ∂x
d Λ = c 1 Λ Λ x Λ ,x Φ = ∂x
d Φ = c 1 Φ Φ x Φ ,
(H.52)
∂Λ∗
∂Φ
∂Λ = c 2 ∂y
d Λ
∂Φ = c 2 ∂y
d Φ
y Λ = ∂y
d Λ = c 2 Λ Λ y Λ ,y Φ = ∂y
d Φ = c 2 Φ Φ y Φ ,
∂Λ
∂Φ
 
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