19-2 Special conic projections based upon the ellipsoid-of-revolution 411
We fix the integration constant:
∆ =
π
2
:
r = f (π/2) = c .
(19.34)
We summarize the general form of the mapping equations:
α
r
=
⎡
⎢
⎣
nΛ
c
2
−
A
2
1 (1 − E
2 )
n
cos ∆
1 − E 2 cos 2 ∆
+
1
2E
ln
1 + E cos ∆
1 − E cos ∆
1/2
⎤
⎥
⎦ .
(19.35)
19-231 Equiareal mapping: the variant of type equidistant and conformal on the reference circle
The projection constant n is fixed by the postulate of an equidistant and conformal mapping on the
reference circle Φ 0 .
Condition on Φ 0 :
A 1 sin ∆ 0 Λ
(1 − E 2 cos 2 ∆ 0 ) 1/2 = f (∆ 0 )nΛ ,
n = cos ∆ 0 ,
A
2
1 sin
2 ∆ 0
1 − E 2 cos 2 ∆ 0
=
c
2
−
A
2
1 (1 − E
2 )
n
cos ∆ 0
1 − E 2 cos 2 ∆ 0
+
1
2E
ln
1 + E cos ∆ 0
1 − E cos ∆ 0
n
2
⇒
c
2 =
A
2
1 (1 + tan
2 ∆ 0 − E
2 )
1 − E 2 cos 2 ∆ 0
+
A
2
1 (1 − E
2 )
2E cos ∆ 0
ln
1 + E cos ∆ 0
1 − E cos ∆ 0
,
(19.36)
1
cos 2 ∆ 0
= 1 + tan
2 ∆ 0
⇒
c
2 = A
2
1
1
cos 2 ∆ 0
+
1 − E
2
2E cos ∆ 0
ln
1 + E cos ∆ 0
1 − E cos ∆ 0
⇒
r = f (∆) =
= A 1
1
cos 2 ∆ 0
−
(1 − E
2 ) cos ∆
cos ∆ 0 (1 − E 2 cos 2 ∆)
+
1 − E
2
2E cos ∆ 0
ln
1 + E cos ∆ 0
1 − E cos ∆ 0
1 − E cos ∆
1 + E cos ∆
1/2
.
(19.37)