17-2 Special mapping equations 389
17-23 Equal area mapping (Albers projection)
The general mapping equations for this type of mappings are derived from the requirement that the
product of the principal stretches equals unity, i. e.
Λ 1 Λ 2 =
nf
R cos Φ
f
R
= 1 ⇒ ff
=
R
2 cos Φ
n
⇒
f df =
R
2
n
cos ΦdΦ
⇓
1
2
f
2 = −
R
2
n
sin Φ +
1
2
c ⇒ f =
−
2R 2
n
sin Φ + c .
(17.39)
For the root to be real for all Φ, the integration constant c should fulfill the inequality c ≥ 2
R
2
n . The
general mapping equations thus are given by (17.40) or (17.41), and the general left principal stretches
are given by (17.42).
α
r
=
⎡
⎣
nΛ
−
2R 2
n sin Φ + c
⎤
⎦ ,
(17.40)
x
y
=
−
2R 2
n
sin Φ + c
cos(nΛ)
sin(nΛ)
,
(17.41)
Λ 1 =
n
−
2R 2
n sin Φ + c
R cos Φ
, Λ 2 =
R cos Φ
n
−
2R 2
n sin Φ + c
.
(17.42)
17-231 Equidistance and conformality on the circle-of-contact, compare with Fig. 17.10
For the reason to map the standard parallel (circle-of-contact) Φ = Φ 0 equidistantly, we claim that
(17.43) holds, with the consequence that – together with the cone constant n = sin Φ 0 – (17.44) is
immediately obtained.
Λ 1 | Φ=Φ 0 =
n
−
2R 2
n sin Φ 0 + c
R cos Φ 0
= 1 ,
(17.43)
c = R
2 (2 + cot
2 Φ 0 ) .
(17.44)
The mapping equations therefore are provided by (17.45) or (17.46). The left principal stretches are
provided by (17.47).
α
r
=
⎡
⎣
nΛ
R
−
2
n sin Φ + cot
2 Φ 0 + 2
⎤
⎦ ,
(17.45)
x
y
= R
−
2
n
sin Φ + cot
2 Φ 0 + 2
cos(nΛ)
sin(nΛ)
,
(17.46)
Λ 1 =
√
−2n sin Φ + n 2 + 1
cos Φ
, Λ 2 =
cos Φ
√ −2n sin Φ + n 2 + 1
.
(17.47)