174 5 “Sphere to tangential plane”: polar (normal) aspect
5-24 Normal perspective mappings
The general normal perspective mapping of the sphere S
2
R of radius R to a tangential plane at the
North Pole, the South Pole, or an equatorial plane is of focal interest in Mathematical Cartography,
in Photogrammetry, in Machine Vision as well as in Aeronautics and Satellite Geodesy. Here, as soon
as we have generated the general parameterized mappings of the perspective type, we introduce more
specific projections: the gnomonic projection, the orthographic projection, and the Lagrange projection.
Based on Figs. 5.10–5.12, we design the elements of a first perspective projection. At first, we locate the
perspective center at O
∗ outside the sphere on the southern axis of symmetry North-Pole–South-Pole.
The perspective center O
∗ is the origin of a bundle of projection lines, in particular, half straights.
Second, we place the projection plane (i) at maximum distance from O
∗ at the North Pole to coincide
with the tangential plane T N S
2
R , (ii) at the center O of the sphere S
2
R as the equatorial plane, and (iii)
at minimum distance from O
∗ at the South Pole to coincide with the tangential plane T N S
2
R . Note
that the projection lines intersect the sphere S
2
R at P , while the projection plane is intersected at p.
The perspective center O
∗ is at distance D from the origin O of the sphere S
2
R or at height H above
S, measured by SO
∗ , such that D = R + H holds.
Question.
Question: “How to find the polar coordinate r = f (∆) in Figs. 5.10–5.12, where ∆ = π/2 − Φ
is the spherical colatitude and Φ is the spherical latitude of the point P ∈ S
2
R ?” Answer:
“Consult the sub-sections that follow, which compactly present the case studies for the
individual geometrical situations.”
Note that in all these cases the perspective ratio r/QP = O
∗ N/O
∗ Q is the fundament for the answer
to the well-posed question.
N
S
r
P
p
O
Q = π(P )
R
R
H
O
∗
R
D
Fig. 5.10. Spherical vertical section, general normal perspective mapping of the sphere to the tangential plane
at the North Pole, projection plane at maximal distance.
5-24 Normal perspective mappings
The general normal perspective mapping of the sphere S
2
R of radius R to a tangential plane at the
North Pole, the South Pole, or an equatorial plane is of focal interest in Mathematical Cartography,
in Photogrammetry, in Machine Vision as well as in Aeronautics and Satellite Geodesy. Here, as soon
as we have generated the general parameterized mappings of the perspective type, we introduce more
specific projections: the gnomonic projection, the orthographic projection, and the Lagrange projection.
Based on Figs. 5.10–5.12, we design the elements of a first perspective projection. At first, we locate the
perspective center at O
∗ outside the sphere on the southern axis of symmetry North-Pole–South-Pole.
The perspective center O
∗ is the origin of a bundle of projection lines, in particular, half straights.
Second, we place the projection plane (i) at maximum distance from O
∗ at the North Pole to coincide
with the tangential plane T N S
2
R , (ii) at the center O of the sphere S
2
R as the equatorial plane, and (iii)
at minimum distance from O
∗ at the South Pole to coincide with the tangential plane T N S
2
R . Note
that the projection lines intersect the sphere S
2
R at P , while the projection plane is intersected at p.
The perspective center O
∗ is at distance D from the origin O of the sphere S
2
R or at height H above
S, measured by SO
∗ , such that D = R + H holds.
Question.
Question: “How to find the polar coordinate r = f (∆) in Figs. 5.10–5.12, where ∆ = π/2 − Φ
is the spherical colatitude and Φ is the spherical latitude of the point P ∈ S
2
R ?” Answer:
“Consult the sub-sections that follow, which compactly present the case studies for the
individual geometrical situations.”
Note that in all these cases the perspective ratio r/QP = O
∗ N/O
∗ Q is the fundament for the answer
to the well-posed question.
N
S
r
P
p
O
Q = π(P )
R
R
H
O
∗
R
D
Fig. 5.10. Spherical vertical section, general normal perspective mapping of the sphere to the tangential plane
at the North Pole, projection plane at maximal distance.