1-10 Two examples: Mercator Projection and Stereographic Projection 71
P l
A 2
Z l
A 1
p l = π(P l )
N l
0 l
S l
0 r
P r
r
r
N r
Z r
S r
p r = π(P r )
π
4
−
φ
2
Fig. 1.27. Left: vertical section of the “left” ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2 , projective geometry of type
central perspective (perspective center S l , projection line S l P l p l , projective plane P
2
N l
). Right: vertical section
of the “right” sphere S
2
r , projective geometry of type central perspective (perspective center S r , projection line
S r P r p r , projective plane P
2
N r ).
Box 1.38 (Projective geometry of type central perspective).
Left projective ratio:
Right projective ratio:
Z l P l
N l p l
=
S l Z l
S l N l
,
√
X 2 + Y 2
p
P 2 + Q 2
=
A 2 + Z
2A 2
.
Z r P r
N r p r
=
S r Z r
S r N r
,
p
x 2 + y 2
p
p 2 + q 2
=
r + z
2r
.
(1.260)
f (Φ):
g(φ):
A 1 cos Φ
p
1 − E 2 sin
2 Φ
1
p
P 2 + Q 2
=
r cos φ
p
p 2 + q 2
=
=
1
2
1 +
√
1 − E 2 sin Φ
p
1 − E 2 sin
2 Φ
!
,
1
2
(1 + sin φ) ,
p
P 2 + Q 2 = 2A 1
cos Φ
p
1 − E 2 sin
2 Φ +
√
1 − E 2 sin Φ
,
p
p 2 + q 2 = 2r
sin
` π
2
− φ
´
1 + cos
` π
2
− φ
´ ,
p
P 2 + Q 2 =
2A 1
√
1 − E 2
×
p
p 2 + q 2 = 2r tan
„
π
4
−
φ
2
«
×
cos Φ
p
1 − E 2 sin
2 Φ/
√
1 − E 2 + sin Φ
,
:= g(φ) .
p
P 2 + Q 2 =
2A 1
√
1 − E 2
×
×
cos Φ
1 + sin Φ
1 + sin Φ
p
1 − E 2 sin
2 Φ/
√
1 − E 2 + sin Φ
,
p
P 2 + Q 2 =
2A 1
√
1 − E 2
tan
„
π
4
−
φ
2
«
×
×
p
(1 + E)(1 − E)(1 + sin Φ)
p
(1 + E sin Φ)(1 − E sin Φ) +
p
(1 + E)(1 − E) sin Φ
=:
=: f (Φ) .
(1.261)