6.9. GEOMETRY OF OBLIQUE PHOTOGRAPHS

Technically, a tilted photograph is a low oblique photograph. In fact, an oblique photograph may be considered as a "vertical" photograph with deliberate tilt of known angle between 3o and 90o. It follows that the geometry of an oblique photograph is similar to that of a vertical photograph and the same principles apply. In an oblique photograph, however, because the tilt is substantial, the combined effects of relief and tilt are intensified and the relationship among the three photo centers (i.e., principal point, nadir, and isocenter) is more apparent.

6.9.1. Geometry of high oblique photographs

Figure 6.30 illustrates the principal plane section of a high oblique photograph and the resulting print. The geometry of a high oblique photograph requires the knowledge of three principal parameters: the focal length of the lens (f), the angle of tilt (q) or its complement, the depression angle (a), and the flying altitude of the aircraft above the datum (HD).

The angle of tilt, q, is the angle between the perpendicular projection through the center of the lens (LO) and the plumb line (LN) as shown in figure 6.30a. The complement angle (90o - q) is the depression angle (a) of the photograph, which is the angle between the optical axis (LO) and the true horizon (Lz). The focal length, f, is the distance between the center of the lens (L), where the photograph was taken from, and the principal point (o) of the photograph. The flying height of the aircraft (HD) is the vertical altitude above the datum, which is equal to the distance LN in figure 6.30a.
 

Figure 6.30. Geometry of an aerial high oblique photograph; (a) the principal plane section and (b) the resulting image.
Other pertinent parameters to the geometry of a high oblique photograph are the true horizon, the apparent horizon, the apparent depression angle, and the dip angle, and shown in figure 6.31.

The true horizon line is the intersection of the horizonal plane containing the camera lens and the plane of the oblique photograph. It is an imaginary line and, therefore, does not appear on the photograph.

The apparent horizon line is the actual line where the earth surface appears to meet with the sky. This line is not imaginary, it is visible on the high oblique photograph and due to the curvature of the earth, its image will be slightly curved on a small scale high oblique photograph.
 

Figure 6.31. Diagram illustrating the true horizon, the apparent horizon, and the dip angle.
The dip angle, d, is the angle measured in the principal plane between the true and the apparent horizons. The dip angle is produced by the height of the aircraft (i.e., camera) above the ground and it increases proportionally as the height of the aircraft. From figure 6.31, if HD is the flying height above the datum, R is the radius of the earth, LZ is the true horizon, and LZ' is the apparent horizon, then, d can be determined as:

where LZ' is determined by the Pythagorean theorem as:

Substituting for LZ',

Since HD is small compared to R, this equation reduces to:

Because atmospheric refraction reduces the value of d, this latter needs to be multiplied by the constant 0.9216 to account for this effect. Further, because d is always small, tand @ d in radians. Converting radians to seconds and substituting the mean earth radius (20.9 x 106 ft) into the equation, equation 6.42 is reduced to:

if HD is expressed in feet, or

if HD is expressed in meters.

As seen from equation 6.43, the dip angle, expressed in minutes, is approximately equal to the square root of the flying height above the ground in feet:

or, when HD is expressed in meters.

Apparent depression angle is the angle, aa, measured in the principal plane, between the optical axis of the camera and the apparent horizon (see figure 6.30a). Since the apparent horizon is visible, and usually traced, on a high oblique photograph, the distance oz' can be measured. Further, because oz' is perpendicular to Lo (figure 6.30), aa can be calculated as:

Due to the curvature of the earth and the flying height of the aircraft above the ground, the apparent horizon falls below the true horizon. Therefore, the apparent depression angle should always be increased by the dip angle to obtain the true depression angle:

Example 6.16

Assume a 152.4-mm focal length camera was used to take an oblique photograph at a flying height of 3000 m above datum. On the tilted photograph, the distance between the principal point and the apparent horizon was measured to be 9.5 cm. Find the true depression angle.

Solution:

From equation 6.44,

and from equation 6.47,

Finally, from equation 6.48, a = aa + d = 33.56 degrees, or 33o33'28"

When the true depression angle is determined, the correct position of the true horizon, in respect to the principal point, may then be calculated as follows:

In figure 6.30a, where oz is the distance between the true horizon (z) and the principal point (o), f is the focal length, and a is the depression angle of the photograph from the true horizon.

At this junction, we should note that the oblique photograph considered in figure 6.30 was taken from an aircraft being nose-up-tail-down. In this case, only the longitudinal tilt (i.e., x-tilt) is assumed to be affecting the geometry of the oblique photograph. However, the lateral tilt, or y-tilt, (aircraft being wing-up-wing-down), as in the case of a vertical photograph, may also have significant effect on the geometry of an oblique photograph. When the y-tilt is significant, the geometry of an oblique photograph is different from that of figure 6.30. Since the y-tilt is usually very slight compared to the x-tilt, it is often insignificant, thus usually ignored.

Because of large tilt angle on high oblique photographs, the nadir point (n) may lie outside the photograph limits at the intersection of the vertical (plumb line) from the camera lens (L) and the plane of the tilted photograph (i.e., the plane passing through o and i in figure 6.30a). Notice that the principal point (o), the nadir (n), and the isocenter (i) are very distinct. Had the photograph been free of tilt, the optical axis would have intersected the photograph plane at the isocenter (i), in which case, the three photo centers would coincide.

Figure 6.30b portrays the diagrammatic representation of the photograph produced by the geometric settings in figure 6.30a. On this diagram, the principal meridian of the photograph transects the four centers of radial displacement: the nadir (n), the isocenter (i), the optical center or principal point (o), and the horizon point (z).

When the oblique photograph is assumed free of y-tilt and either the tilt angle (q) or the depression angle a is known, the position of each of the four points (o, n, i, and z) on the principal meridian can be determined. Conversely, if either z, i, or n can be located with respect to o on the photograph, the tilt angle (q) or it complement (a = 90o - q) may be computed readily from the same formulas . This is most often possible in the case of high oblique photographs where the position of the apparent horizon is pictured on the photograph.

Once the position of the true horizon is located on the principal meridian (equation 6.49), the position of the other three points (o, n, and i) can readily be determined. Since the true horizon was determined in respect to the principal point, the same formula (equation 6.49) is used to locate the principal point in respect to the true horizon (z).

Similarly, the positions of the nadir point (n) and the isocenter (i) from the principal point (o) may be computed as:

where no is the distance between the nadir and the principal point, io is the distance between the isocenter and the principal point, and q is the angle of tilt of the photograph.

Example 6.17

A high oblique photograph was taken at a flying height of 3000 m above datum with a 152.4-mm focal length camera. The photo distance between the principal point and the apparent horizon, measured along the principal line, is 7.50 cm. Find the true depression angle and the positions of the true horizon, the nadir, and the isocenter with respect to the principal point.

Solution:

The position of the true horizon with respect to the principal point is determined by the depression angle, which is the sum of the apparent depression angle and the dip angle.

The apparent depression angle is obtained by equation 6.47:

and the dip angle is obtained by equation 6.44:

The true depression angle is then, 26o 12' 11" + 1o 37' 13" = 27o 49' 24" or 27.88 degrees

Using equation 6.49, the position of the true horizon, with respect to the principal point is:

oz = 152.4 mm tan 27.88 = 80.63 mm

and the positions of the nadir and the isocenter are obtained from equations 6.50 and 6.51, respectively as:

on = 152.4 mm tan (90o - 27.88) = 288.04 mm

oi = 152.4 mm tan [(90o - 27.88)/2] = 91.78 mm

Because of the exaggerated perspective distortions on an oblique photograph, figure 6.30b shows that equidistant lines drawn perpendicularly to the principal meridian appear to be imaged closer and closer as their distance from the nadir increases. Similarly, lines which are actually parallel to the principal meridian appear to converge at point z on the true horizon. Consequently, the size of equal squares of a grid constructed by these lines on the ground would appear smaller and smaller as they get farther from the nadir. The techniques of constructing, on the photograph, the equidistant lines and those converging at the true horizon point are described by Spurr (1960).

6.9.2. Geometry of low oblique photographs

As seen in the case of a high oblique photograph, the development of the geometry of an oblique photograph depends on the visibility of the apparent horizon on the photograph. The lack of a horizon on a low oblique photograph is the major difference between this type of photograph and a high oblique photograph. Because of the absence of a horizon on a low oblique photograph, the determination of the former becomes dependable on the type of features imaged on the photograph. Two possibilities may be used to locate the position of the horizon on a low oblique photograph.

First method: If ground features of horizontal parallel lines (e.g., limits of parcels) are clearly visible on the photograph, such as in figure 6.32, the projection of these lines will intersect at their corresponding so-called vanishing points on the true horizon. From these vanishing points, the location of the true horizon point (z) can be constructed. Once the true horizon point is positioned, the same geometry developed for a high oblique photograph applies. The depression angle can be determined by measuring the distance, oz, between the principal point and the horizon point and applying equation 6.47.

Second method: This method depends on the presence of perfectly vertical tall objects (e.g., towers, trees, poles, buildings) on the oblique photograph (figure 6.33). If these objects are clearly visible, the vertical lines in perspective will converge at the nadir point. Once the nadir point is located, the distance, no, between the nadir and the principal point can be measured and the depression angle can be computed from equation 6.49 (a = 90o - q).
 

6.32. Determination of the true horizon from horizontal parallel lines.
6.33. Determination of nadir point by convergence of vertical lines.
6.8 Combined Effects of Relief and Tilt | 6.9 Geometry of Oblique Photographs | References