298 13 “Sphere to cylinder”: pseudo-cylindrical projections
13-23 Parabolic pseudo-cylindrical mapping (J. E. E. Craster), compare with Fig. 13.3
This mapping is defined in such a way that the meridians except the central meridian, which is
a straight line, are equally spaced parabolas. Parallels are unequally spaced straight lines, farthest
apart near the equator. The mapping equations are defined by (13.26). The left Jacobi matrix is given
by (13.27) and the left Cauchy–Green matrix is given by (13.28) (G r = I 2 ). As can be seen from
Fig. 13.3, map distortion is severe near outer meridians at high latitudes.
x =
=
3
π RΛ(2 cos
2Φ
3 − 1) ,
y =
=
√
3πR sin
Φ
3 ,
(13.26)
J l =
= R
3
π
2 cos
2Φ
3 − 1 −
4
3 Λ sin
2Φ
3
0
π
3 cos
Φ
3
,
(13.27)
C l = J
∗
l G r J l =
=
R
2
3π
9
1 − 2 cos
2Φ
3
2
12Λ sin
2Φ
3
1 − 2 cos
2Φ
3
12Λ sin
2Φ
3
1 − 2 cos
2Φ
3
16Λ
2 sin
2 2Φ
3 + π
2 cos
2 Φ
3
.
(13.28)
Fig. 13.3. Equal area pseudo-cylindrical mapping. Craster projection.