cally. 
  
  
Figure 3. The distorted two-point perspective 
to be rectified by the first technique de- 
scribed in this paper. 
original scene was expressed in the simplest 
grid patterns, the equatorial and the polar 
grids. 
The first fisheye photograph to be corrected 
here was taken with the film plane parallel to 
the vertical straight lines that represented 
the corners of the building, setting up a dis- 
torted form of two-point perspective (Fig. 3). 
The photograph was also set up so that two of 
the building's sides were visible. In the re- 
sulting distorted view, the vertical edges of 
the building obeyed the vertical arc-shaped 
curves of the equatorial grid and were read as 
so many degrees right or left of the central 
meridian of the grid, and were transferred into 
the bottom semicircle shown in a newly devised 
graphical setup (Fig. 4). The upper and lower 
limits of each vertical edge of the building 
were read off the horizontal arc-shaped curves 
of the same grid, and were transferred to the 
upper semicircle shown in Fig. 4. The various 
angles were extended to intersect two lines 
drawn tangent to the midpoint of the semicir- 
cles; these two tangent lines represent the 
overhead and side views of a plane in the orig- 
inal scene. New extensions perpendicular from 
the plane lines serve to locate the corners of 
the corrected two-point perspective view. If 
any angles had been read below the horizon, 
then the projection planes would have had to 
have been offset by equal distances from the 
semicircles to allow the construction of the 
corrected perspective in a clear space. 
Rectifying this corrected perspective view is 
an application of standard techniques already 
described in any of a variety of references 
(McNeil, 1954; Gracie et. al., 1967; Fry, 1969; 
Wolf, 1974; Kelley, 1978-1983; Busby, 1981; 
Wiliamson and Brill, 1987; and Brill and Wil- 
325 
liamson, 1987--to name but a few). Standard 
mechanical and architectural drafting textbooks 
also describe some of these methods. The rec- 
tification of the corrected perspective yielded 
an orthographic view of the top of the building 
demonstrating a length-width ratio of 3.6:1--a 
match to the in situ measurements of the build- 
ing (67.7 m x 18.6 m,or 222 feet x 61 feet). 
The second fisheye photograph to be corrected 
here was taken with the film plane parallel to 
one full side of the test building, setting up 
a distorted form of one-point perspective (Fig. 
5). In this distorted view, both the vertical 
and horizontal edges of the building followed 
the arc-shaped curves of the same equatorial 
grid used in the previous example. Because 
both sets of lines were interpretable off the 
same grid, the corrected one-point perspective 
involves the same readings and plotting of an- 
gles as before, but there are fewer steps to get 
to the correction (Fig. 6). In this particular 
instance, the corrected one-point perspective 
happens to be same view as an orthographic 
drawing of that side of building. 
OTHER WAYS TO DO THE SAME CORRECTIONS 
Using the arranged photographs in the two ex- 
amples just described is a convenient and 
simple proof-of-concept for the possibility of 
correcting curvilinear perspective. In reali- 
ty, the building could have been imaged at any 
angle so long as the building remained within 
the 180-degree field of view of the fisheye 
lens. In such an instance, the proper oblique 
grid for each set of lines of the distorted 
building could have been used to produce a cor- 
rected three-point perspective drawing. Figure 
7 shows the agreement of one oblique grid to the 
curved lines of the same image previously used 
as Fig. 3. Using this oblique grid, in con- 
junction with the equatorial grid used earlier, 
could have allowed the double-correction of the 
building using only one photograph. 
USES FOR FISHEYE PHOTOGRAPHS 
This author's personal viewpoint is that fish- 
eye photography represents an under-exploited 
resource that could be tasked to do more work 
with fewer images than are required by conven- 
tional photography. There seems to be no 
reason why fisheye photographs could not pro- 
vide more coverage of crime and accident scenes 
with fewer images, and produce maps or other 
drawings of the same accuracy as a greater num- 
ber of conventional photographs. The polar 
grids could be used to exploit fisheye views 
taken in either zenith or nadir positions to 
yield maps in close quarters. Any of the ob- 
lique grids might also be applied to mapping 
within cramped spaces. It is only conjectural, 
but the curvilinear perspective of the fisheye 
view might be directly related to the curved- 
line imagery of slightly longer focal-length 
wide-angle lenses as well as the apparent 
straight-line imagery of really long focal- 
length lenses. However, any such reality for