184 5 “Sphere to tangential plane”: polar (normal) aspect
√
1 + x
1/
√
1 + x
x
x
1.0
1.5
2.0
2.5
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
1
2
3
4
5
0
1
2
3
4
5
Fig. 5.15. M
aximal extension of a normal central perspective of the sphere, the functions
√
1 + x and
1/
√
1 + x, domain 0 ≤ x ≤ 5, x := 2R/H.
Such a discussion motivates the construction of a minimal atlas from the setup of a normal central
perspective as follows. Consider the two charts (i) central perspective projection from a southern
perspective center to a projection plane at the North Pole and (ii) central perspective projection from
a northern perspective center to a projection plane at the South Pole. The union of the two charts
covers the sphere S
2
R completely. The two charts constitute a minimal atlas. While Fig. 5.10 illustrates
the first chart (O
∗ south on the NS line, projection plane at the North Pole), in contrast, Fig. 5.16
illustrates the second chart (O
∗ north on the SN line, projection plane at the South Pole).
Table 5.1. M
aximal extension of a normal central perspective of the sphere, the functions
√
1 + x and
1/
√
1 + x, domain 0 ≤ x ≤ 5, x := 2R/H.
x
√
1 + x 1/
√
1 + x
0
.0
1.0 0
0
1.0 0
0
0
.5
1.225
0
.816
1.0
1.4 14
0
.7 0
7
1.5
1.581
0
.6 3
2
2.0
1.7 3
2
0
.57 7
2.5
1.87 1
0
.53 5
3
.0
2.0 0
0
0
.50 0
3
.5
2.121
0
.4 7
1
4
.0
2.23 6
0
.4 4
7
4
.5
2.3 4
5
0
.4 26
5.0
2.4 4
9
0
.4 0
8
√
1 + x
1/
√
1 + x
x
x
1.0
1.5
2.0
2.5
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
1
2
3
4
5
0
1
2
3
4
5
Fig. 5.15. M
aximal extension of a normal central perspective of the sphere, the functions
√
1 + x and
1/
√
1 + x, domain 0 ≤ x ≤ 5, x := 2R/H.
Such a discussion motivates the construction of a minimal atlas from the setup of a normal central
perspective as follows. Consider the two charts (i) central perspective projection from a southern
perspective center to a projection plane at the North Pole and (ii) central perspective projection from
a northern perspective center to a projection plane at the South Pole. The union of the two charts
covers the sphere S
2
R completely. The two charts constitute a minimal atlas. While Fig. 5.10 illustrates
the first chart (O
∗ south on the NS line, projection plane at the North Pole), in contrast, Fig. 5.16
illustrates the second chart (O
∗ north on the SN line, projection plane at the South Pole).
Table 5.1. M
aximal extension of a normal central perspective of the sphere, the functions
√
1 + x and
1/
√
1 + x, domain 0 ≤ x ≤ 5, x := 2R/H.
x
√
1 + x 1/
√
1 + x
0
.0
1.0 0
0
1.0 0
0
0
.5
1.225
0
.816
1.0
1.4 14
0
.7 0
7
1.5
1.581
0
.6 3
2
2.0
1.7 3
2
0
.57 7
2.5
1.87 1
0
.53 5
3
.0
2.0 0
0
0
.50 0
3
.5
2.121
0
.4 7
1
4
.0
2.23 6
0
.4 4
7
4
.5
2.3 4
5
0
.4 26
5.0
2.4 4
9
0
.4 0
8