“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 207

Fig. 5.14. Line-of-sight, normal perspective mapping of the sphere to the tangential plane at the South Pole,
projection plane at minimal distance

Question: “What is the line-of-sight or the line-of-contact
and how can we compute the spherical latitude Φ r of the
line-of-contact or the maximal radial coordinate r max ?”
Answer 1: “The normal central projection O ∗ →

T N S

2 R or

O ∗ →

2 R is restricted to points inside the circular cone

T S S

2
Q lr P r

2
P l Q lr

C . Indeed, the projection line, which contacts

the sphere tangentially, restricts the domain of points of

or C

2 R

S

2 R . The radius Q lr P r
or P l Q lr determines the circular cone. Its related bundle of
projection lines constitutes the characteristic circular cone-
of-contact. The line-of-contact is the circle

2 R or T S S

which can be mapped to T N S

1
R cos Φ r

of radius
R cos Φ r . Its trace P l Q lr P r is illustrated in Figs. 5.13 and
5.14 , respectively.” Answer 2: “Let be given the distance
O ∗ O

S

O ∗ and the origin

2

R ,
which is called D , or alternatively the spherical height H
of the perspective center

of the perspective center O of S

O ∗ relative to S. Then the crit-
ical spherical latitude Φ r can be computed as outlined in
Box 5.13 .If O ∗ is placed south on the line NS, then the
critical value is determined by sin |Φ r | = R/D , regardless
whether the projection plane is located at the North Pole
or at the South Pole.”