لخّصلي

Geometry of Vertical Photographs
As described in Chap. 1, photographs taken from an aircraft with the optical axis of the camera
vertical or as nearly vertical as possible are called vertical photographs. If the optical axis is exactly
vertical, the resulting photograph is termed truly vertical. In this chapter, equations are developed
assuming truly vertical photographs. In spite of precautions taken to keep the camera axis vertical,
small tilts are invariably present. For photos intended to be vertical, however, tilts are usually less
than 1° and rarely exceed 3°. Photographs containing these small unintentional tilts are called nearvertical
or tilted photographs, and for many practical purposes these photos may be analyzed using
the relatively simple “truly vertical” equations of this chapter without serious error.
In this chapter, besides assuming truly vertical photographs, other assumptions are that the photo
coordinate axis system has its origin at the photographic principal point and that all photo coordinates
have been corrected for shrinkage, lens distortion, and atmospheric refraction distortion.
Figure 6-1 illustrates the geometry of a vertical photograph taken from an exposure station L. The
negative, which is a reversal in both tone and geometry of the object space, is situated a distance equal
to the focal length (distance o′L in Fig. 6-1) above the rear nodal point of the camera lens. The positive
may be obtained by direct emulsion-to-emulsion “contact printing” with the negative. This process
produces a reversal of tone and geometry from the negative, and therefore the tone and geometry of
the positive are exactly the same as those of the object space. Geometrically the plane of a contactprinted
positive is situated a distance equal to the focal length (distance o′L in Fig. 6-1) below the
front nodal point of the camera lens. The same is true for an image obtained with a frame-type digital
camera. The reversal in geometry from object space to negative is readily seen in Fig. 6-1 by
comparing the positions of object points A, B, C, and D with their corresponding negative positions a′,
b′, c′, and d′. The correspondence of the geometry of the object space and the positive is also readily
apparent. The photographic coordinate axes x and y, as described in Chap. 4, are shown on the positive
of Fig. 6-1.
FIGURE 6-1 The geometry of a vertical photograph.
6-2 Scale
Map scale is ordinarily interpreted as the ratio of a map distance to the corresponding distance on the
ground. In a similar manner, the scale of a photograph is the ratio of a distance on the photo to the
corresponding distance on the ground. Due to the nature of map projections, map scale is not
influenced by terrain variations. A vertical aerial photograph, however, is a perspective projection,
and as will be demonstrated in this chapter, its scale varies with variations in terrain elevation.
Scales may be represented as unit equivalents, unit fractions, dimensionless representative
fractions, or dimensionless ratios. If, for example, 1 inch (in) on a map or photo represents 1000 ft
(12,000 in) on the ground, the scale expressed in the aforementioned four ways is
1. Unit equivalents: 1 in = 1000 ft
2. Unit fraction: 1 in/1000 ft
3. Dimensionless representative fraction: 1/12,000
4. Dimensionless ratio: 1:12,000
By convention, the first term in a scale expression is always chosen as 1. It is helpful to remember that
a large number in a scale expression denotes a small scale, and vice versa; for example, 1:1000 is a
larger scale than 1:5000.
6-3 Scale of a Vertical Photograph Over Flat Terrain
Figure 6-2 shows the side view of a vertical photograph taken over flat terrain. Since measurements
are normally taken from photo positives rather than negatives, the negative has been excluded from
this and other figures that follow in this text. The scale of a vertical photograph over flat terrain is
simply the ratio of photo distance ab to corresponding ground distance AB. That scale may be
expressed in terms of camera focal length f and flying height above ground H′ by equating similar
triangles Lab and LAB as follows:
FIGURE 6-2 Two-dimensional view of a vertical photograph taken over flat terrain.
(6-1)
From Eq. (6-1) it is seen that the scale of a vertical photo is directly proportional to camera focal
length (image distance) and inversely proportional to flying height above ground (object distance).
Example 6-1
A vertical aerial photograph is taken over flat terrain with a 152.4 mm-focal-length camera from an
altitude of 1830 m above ground. What is the photo scale?
Solution By Eq. (6-1),
Note the use of the overbar in the solution of Example 6-1 to designate significant figures, as
discussed in Sec. A-3.
6-4 Scale of a Vertical Photograph Over Variable Terrain
If the photographed terrain varies in elevation, then the object distance—or the denominator of Eq. (6-
1)—will also be variable and the photo scale will likewise vary. For any given vertical photo scale
increases with increasing terrain elevation and decreases with decreasing terrain elevation.
Suppose a vertical aerial photograph is taken over variable terrain from exposure station L of Fig.
6-3. Ground points A and B are imaged on the positive at a and b, respectively. Photographic scale at
h, the elevation of points A and B, is equal to the ratio of photo distance ab to ground distance AB. By
similar triangles Lab and LAB, an expression for photo scale SAB is
FIGURE 6-3 Scale of a vertical photograph over variable terrain.
(a)
Also, by similar triangles LOAA and Loa,
(b)
Substituting Eq. (b) into Eq. (a) gives
(c)
Considering line AB to be infinitesimal, we see that Eq. (c) reduces to an expression of photo scale at a
point. In general, by dropping subscripts, the scale at any point whose elevation above datum is h may
be expressed as
(6-2)
In Eq. (6-2), the denominator H - h is the object distance. In this equation as in Eq. (6-1), scale of a
vertical photograph is seen to be simply the ratio of image distance to object distance. The shorter the
object distance (the closer the terrain to the camera), the greater the photo scale, and vice versa. For
vertical photographs taken over variable terrain, there are an infinite number of different scales. This
is one of the principal differences between a photograph and a map.
6-5 Average Photo Scale
It is often convenient and desirable to use an average scale to define the overall mean scale of a
vertical photograph taken over variable terrain. Average scale is the scale at the average elevation of
the terrain covered by a particular photograph and is expressed as
(6-3)
When an average scale is used, it must be understood that it is exact only at those points that lie at
average elevation, and it is an approximate scale for all other areas of the photograph.
Example 6-2
Suppose that highest terrain h1, average terrain havg, and lowest terrain h2 of Fig. 6-3 are 610, 460, and
310 m above mean sea level, respectively. Calculate the maximum scale, minimum scale, and average
scale if the flying height above mean sea level is m and the camera focal length is 152.4 mm.
Solution By Eq. (6-2) (maximum scale occurs at maximum elevation),
and (minimum scale occurs at minimum elevation)
By Eq. (6-3)
In each of Eqs. (6-1), (6-2), and (6-3), it is noted that flying height appears in the denominator.
Thus, for a camera of a given focal length, if flying height increases, object distance H - h increases
and scale decreases. Figures 6-4a through d illustrate this principle vividly. Each of these vertical
photos was exposed using the very same 23-cm format and 152-mm-focal-length camera. The photo
of Fig. 6-4a had a flying height of 460 m above ground, resulting in an average photo scale of 1:3000.
The photos of Fig. 6-4b, c, and d had flying heights above average ground of 910 m, 1830 m, and 3660
m, respectively, producing average photo scales of 1:6000, 1:12,000, and 1:24,000, respectively