Related Calculations: Let A denote a point at an elevation h above the datum,
let B denote a point that lies vertically below A and in the datum plane, and let a and b de-
note the images of A and B, respectively. As Fig. 30 shows, a and b lie on a straight line
that passes through o, which is called a radial line. The distance d = ba is the displace-
ment of the image of A resulting from its elevation above the datum, and it is termed the
relief displacement of A. Thus, the relief displacement of a point is radially outward if that
point lies above datum and radially inward if it lies below datum. From above, ob/oa =
(H- K)IH, where H= flying height above datum. Then d = oa - ob = (oa)h/H.
DETERMINING GROUND DISTANCE
BY TILTED PHOTOGRAPH
Two points A and B are located on the ground at elevations of 180 and 13Om, respective-
ly, above sea level. Points A and B have images a and b, respectively, on an aerial photo-
graph, and the coordinates of the images are xa = +40.63 mm, ya = -73.72 mm, xb =
-78.74 mm, andyfe = +20.32 mm. The focal length is 153.6 mm, and the flying height is
2360 m above sea level. By use of ground control points, it was established that the pho-
tograph has a tilt of 2°54' and a swing of 162°. Determine the distance between A and B.
Calculation Procedure:
- Compute the transformed coordinates of the images
Refer to Fig. 31, where L again denotes the front nodal point of the lens and o denotes the
principal point. A photograph is said to be tilted, or near vertical, if by inadvertence the
optical axis of the lens is displaced slightly from the vertical at the time of exposure. The
tilt t is the angle between the optical axis and the vertical. The principal plane is the verti-
cal plane through the optical axis. Since the plane of the photograph is normal to the opti-
cal axis, it is normal to the principal plane. Therefore, Fig. 3 Ia is an edge view of the
plane of the photograph. Moreover, the angle between the plane of the photograph and the
horizontal equals the tilt. In Fig. 31, A is a point on the ground and a is its image. Line AQ
is normal to the principal plane, Q lies in that plane, and q is the image of Q.
Consider the vertical line through L. The points n and Af at which this line intersects
the plane of the photograph and the ground are called the nadir point and ground nadir
point, respectively. The line of intersection of the principal plane and the plane of the
photograph, which is line no prolonged, is termed the principal line. Now consider the
vertical plane through o parallel to the line of flight. In the photograph, the x axis is placed
on the line at which this vertical plane intersects the plane of the photograph, with jc val-
ues increasing in the direction of flight. The y axis is normal to the x axis, and the origin
lies at o. The swing s is the angle in the plane of the photograph, measured in a clockwise
direction, between the positive side of the y axis and the radial line extending from o to n.
Transform the x and y axes in this manner: First, rotate the axes in a counterclockwise
direction until the y axis lies on the principal line with its positive side on the upward side
of the photograph; then displace the origin from o to n. Let jc' andy denote, respectively,
the axes to which the x andy axes have been transformed. The x' axis is horizontal. Let 6
denote the angle through which the axes are rotated in the first step of the transformation.
From Fig. 3Ib 9 O= 180°-s.
The transformed coordinates of a point in the plane of the photograph are x' = x cos 6
+y sin 6; y' = -x sin 6 +y cos O +/tan t. In this case, t = 2°54' and O = 180° - 162° = 18°.
