13-2 Special mapping equations 295
13-21 Sinusoidal pseudo-cylindrical mapping (J. Cossin 1570, N. Sanson 1650,
J. Flamsteed 1646–1719), compare with Fig. 13.1
The mapping equations (13.12) are derived from (13.1) in connection with (13.8) and the special
instruction f (Φ) = Φ. The left Jacobi matrix is given by (13.13) and the left Cauchy–Green matrix
(G r = I 2 ) by (13.14). The left principal stretches are defined by (13.15). The structure of the coordinate
lines are defined by (13.16).
x
y
= R
Λ cos Φ
Φ
,
(13.12)
J l = R
cos Φ −Λ sin Φ
0
1
,
(13.13)
C l = J
∗
l G r J l = R
2
cos
2 Φ
−Λ sin Φ cos Φ
−Λ sin Φ cos Φ Λ
2 sin
2 Φ + 1
,
(13.14)
(Λ S ) 1,2 = ±
√
2
2
2 + Λ 2 sin
2 Φ + Λ sin Φ
4 + Λ 2 sin
2 Φ ,
(Λ S ) 3,4 = ±
√
2
2
2 + Λ 2 sin
2 Φ − Λ sin Φ
4 + Λ 2 sin
2 Φ ,
(13.15)
Φ =
y
R
⇒ x = RΛ cos
y
R
,
cos Φ =
x
RΛ
⇒ y = R arccos
x
RΛ
.
(13.16)
Fig. 13.1. Equal area pseudo-cylindrical mapping. Sanson–Flamsteed projection.