612 H Generalized Hammer projection
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
∗ , Φ
∗ ) , y = c 2 y
∗ (Λ
∗ , Φ
∗ ) ,
(H.47)
Λ
∗ (Λ) , Φ
∗ (Φ) ,
(H.48)
subject to
A 1∗ = A 1 , A 2∗ = A 2 , E
∗ = E ,
(H.49)
Λ
∗
Λ =
dΛ
∗
dΛ
= c 3 ,
(H.50)
Φ
∗
Φ =
dΦ
∗
dΦ
= c 4
(1 − E
2 sin
2 Φ
∗ )
2
(1 − E 2 sin
2 Φ) 2
cos Φ
cos Φ ∗ .
(H.51)
The left Cauchy–Green deformation tensor:
x Λ =
∂x
∂Λ
= c 1
∂x
∗
∂Λ ∗
dΛ
∗
dΛ
= c 1 Λ
∗
Λ x
∗
Λ ∗ , x Φ =
∂x
∂Φ
= c 1
∂x
∗
∂Φ ∗
dΦ
∗
dΦ
= c 1 Φ
∗
Φ x
∗
Φ ∗ ,
y Λ =
∂y
∂Λ
= c 2
∂y
∗
∂Λ ∗
dΛ
∗
dΛ
= c 2 Λ
∗
Λ y
∗
Λ ∗ , y Φ =
∂y
∂Φ
= c 2
∂y
∗
∂Φ ∗
dΦ
∗
dΦ
= c 2 Φ
∗
Φ y
∗
Φ ∗ ,
(H.52)
ds
2 = dx
2 + dy
2 =
= (x
2
Λ + y
2
Λ )dΛ
2 + 2(x Λ x Φ + y Λ y Φ )dΛdΦ + (x
2
Φ + y
2
Φ )dΦ
2 =
2
X
A,B=1
c AB dU
A dU
B ,
(H.53)
c 11 = Λ
∗
2
Λ (c
2
1 x
∗
2
Λ ∗ + c
2
2 y
∗
2
Λ ∗ ) , c 12 = Λ
∗
Λ Φ
∗
Φ (c
2
1 x
∗
Λ ∗ x
∗
Φ ∗ + c
2
2 y
∗
Λ ∗ y
∗
Φ ∗ ) , c 22 = Φ
∗
2
Φ (c
2
1 x
∗
2
Φ ∗ + c
2
2 y
∗
2
Φ ∗ ) , (H.54)
p
det [C l ] =
q
c 11 c 22 − c
2
12 = c 1 c 2 Λ
∗
Λ Φ
∗
Φ (x
∗
Λ ∗ y
∗
Φ ∗ − x
∗
Φ ∗ y
∗
Λ ∗ ) .
(H.55)
The postulate of an equiareal mapping:
p
det [C l ] =
p
det [G l ] ⇔ c 1 c 2 Λ
∗
Λ Φ
∗
Φ (x
∗
Λ ∗ y
∗
Φ ∗ − x
∗
Φ ∗ y
∗
Λ ∗ ) =
A
2 (1 − E
2 ) cos Φ
(1 − E 2 sin
2 Φ) 2 ,
(H.56)
x
∗
Λ ∗ y
∗
Φ ∗ − x
∗
Φ ∗ y
∗
Λ ∗ =
q
det [C
∗
l ] =
q
det [G
∗
l ] ,
(H.57)
x
∗
Λ ∗ y
∗
Φ ∗ − x
∗
Φ ∗ y
∗
Λ ∗ =
A
2
1 (1 − E
2 ) cos Φ
∗
(1 − E 2 sin
2 Φ ∗ ) 2 ,
(H.58)
p
det [C l ]
p
det [G l ]
= 1 ⇔ Λ
∗
Λ = c 3 c 1 c 2 Λ
∗
Λ Φ
∗
Φ
(1 − E
2 sin
2 Φ)
2
(1 − E 2 sin
2 Φ ∗ ) 2
cos Φ
∗
cos Φ
= 1
⇔
c 1 c 2 c 3 Φ
∗
Φ
(1 − E
2 sin
2 Φ)
2
(1 − E 2 sin
2 Φ ∗ ) 2
cos Φ
∗
cos Φ
= 1 , Φ
∗
Φ = c 4
(1 − E
2 sin
2 Φ
∗ )
2
(1 − E 2 sin
2 Φ) 2
cos Φ
cos Φ ∗
⇔
(H.59)
c 1 c 2 c 3 c 4 = 1 .
(H.60)