240 8 “Ellipsoid-of-revolution to tangential plane”
8-3 Perspective mapping equations
Setting up perspective mappings “ellipsoid-of-revolution to plane”, the fundamental perspective graph,
Space Photos.
In this section, we intend to present various perspective mappings from the ellipsoid-of-revolution to
the tangential plane, placing the perspective center arbitrarily. Let the position P c be on the top
of the ellipsoid-of-revolution. Furthermore, let us use the orthogonal projection to locate the point
P 0 = p 0 at minimal distance or maximal distance, namely X c − X 0 = min or X c − X 0 = max.
Alternatively, we can take advantage of an orthogonal projection of the ellipsoid-of-revolution to the
sphere, which passes the center O of the ellipsoid-of-revolution. The three variants of the special
perspective mappings “ellipsoid-of-revolution to plane” are illustrated by Figs. 8.8, 8.7, and 8.6.
Technical
aside.
Note that the perspective mappings from the ellipsoid-of-revolution to the tangential plane
are applied to map points-in-space to the tangential planes of the ellipsoid-of-revolution.
Examples are visions from a tower or from an airplane and from an Earth satellite by eye
or by a camera. A special example are images of TV cameras showing clouds – important
information needed for weather reports.
For our introduction, we treat only the case of the mapping of minimal distance. The final mapping
equations, given the coordinates of perspective center
Λ 0 , Φ 0 , H 0
to the plane which is located at
minimal distance from the perspective center, are presented in Box 8.8 in terms of the coordinates
(x
∗ , y
∗ ) p in the tangential plane: see (8.76) and (8.77).
P c
p = π(P max )
P max
p = π(P min )
P min
Fig. 8.6. Perspective mappings of a perspective center P c to the plane which passes the center O of the
ellipsoid-of-revolution E
2
A 1 ,A 2 .
8-3 Perspective mapping equations
Setting up perspective mappings “ellipsoid-of-revolution to plane”, the fundamental perspective graph,
Space Photos.
In this section, we intend to present various perspective mappings from the ellipsoid-of-revolution to
the tangential plane, placing the perspective center arbitrarily. Let the position P c be on the top
of the ellipsoid-of-revolution. Furthermore, let us use the orthogonal projection to locate the point
P 0 = p 0 at minimal distance or maximal distance, namely X c − X 0 = min or X c − X 0 = max.
Alternatively, we can take advantage of an orthogonal projection of the ellipsoid-of-revolution to the
sphere, which passes the center O of the ellipsoid-of-revolution. The three variants of the special
perspective mappings “ellipsoid-of-revolution to plane” are illustrated by Figs. 8.8, 8.7, and 8.6.
Technical
aside.
Note that the perspective mappings from the ellipsoid-of-revolution to the tangential plane
are applied to map points-in-space to the tangential planes of the ellipsoid-of-revolution.
Examples are visions from a tower or from an airplane and from an Earth satellite by eye
or by a camera. A special example are images of TV cameras showing clouds – important
information needed for weather reports.
For our introduction, we treat only the case of the mapping of minimal distance. The final mapping
equations, given the coordinates of perspective center
Λ 0 , Φ 0 , H 0
to the plane which is located at
minimal distance from the perspective center, are presented in Box 8.8 in terms of the coordinates
(x
∗ , y
∗ ) p in the tangential plane: see (8.76) and (8.77).
P c
p = π(P max )
P max
p = π(P min )
P min
Fig. 8.6. Perspective mappings of a perspective center P c to the plane which passes the center O of the
ellipsoid-of-revolution E
2
A 1 ,A 2 .