F. Parallax
1. StereoPhoto Geometry
The basic geometry of an overlapping photo pair is shown in Figure F1. Each point in the overlap area appears on both photographs. They appear in different locations on both photos; this is what allows us to see the image in 3D.
Figure F1 StereoPhoto Geometry 
Except for stricly photo interpretation, we would also like to use the geometry to determing a point's numeric lcation. To do that, we need to undertand and use parallax.
2. Definition
Remember parallax? When using a telescoped instrument, parallax occurs when the sighted object and crosshairs don't come to exact foucs on the back of the observer's eye, Figure F2. Moving the head slightly will cause one image to move across the other.
(a) Initial sight 
(b) Eye slightly shifted sideways 
Figure F2 Parallax 
We don't want parallax when using a telescope because it causes sighting error BUT in stereophotography, parallax helps us determine point distances. Parallax is the amount a point moves between two photos.
Figure F3 shows two points at different elevations appearing on two successive photos.
Figure F3 Successive Photos 
The parallax for each point can be compared by overlaying the photos, Figure F4.
Figure F4 Parallax Distances 
From Figures F3 and F4 it is apparent that parallax is:
 a measurable quantity
 related to elevation: the higher the point, the larger its parallax.
Since parallax can be measured, elevations can be determined provided sufficient support information is available.
3. Measuring Parallax
There are two ways to measure parallax depending on object type and accuracy: monoscopic and stereoscopic.
a. Monoscopic
Parallax direction is parallel with the flight direction. Since the photo x axis is in the direction of flight, parallax can be computed from x coordinates, Figure F5 and Equation F1.
Figure F5 Parallax Direction 
The coordinate system should be based on the flight path since the images shift in that direction (see Chapter D Section 4 of this topic). However unless there is severe course change between photos, using the fiducial system on both photos should be sufficient for most surveying applications.
To visualize parallax distances, Figure F6 shows the left and right images of points a and b on a single diagram.
Figure F6 Parallax determination 
Equation F1 
In order to use the monoscopic method, the image point(s) must be distinct on both photographs. Is is extremely difficult, if not impossible, to determine parallax for points in open fields, on water, etc. For those and similar situations, the stereoscopic method must be used.
b. Stereoscopic
In the stereoscopic method parallax is measured in the 3D view using a floating mark. A floating mark is an artificial object added to both photos so it appears in the 3D view. As the two marks are moved closer together or further apart, it will appear to float higher or lower in the model.
Figure F7 shows two floating marks on a stereogram.

In the 3D view, they appear at different elevations because of their different parallaxes.
The floating marks are etched on glass plates attached to a measurement scale which allows their spacing to be changed. The device is called a parallax bar, Figure F8.
Figure F8 Parallax Bar 
The glass plates are placed on the images, Figure F9, and their spacing adjusted until the floating mark "sits" on the object in the 3D model. That places the individual marks at correct parallax distance for the elevation.
Figure F9 Using a Parallax Bar 
The parallax distance is determined using Equation F2.
Equation F2  
C: Parallax bar constant 
The constant is determined by using Equation F1 to determine the parallax of a point, then measuring it with the parallax bar, Equation F3.
Equation F3 
The advantage of a parallax bar it it can be used on indistinct as well as distinct points. You can float the point until it sits on the ground. then read the scale to determine its parallax.
4. Parallax Equations
a. Ground Coordinates
The ground coordinate system is similar to that of a single vertical photo. The origin and axes directions are defined by the left photo, Figure F10, using the actual flight direction.
Figure F10 Ground Coordinate System 
Coordinates and elevation of ground points can be determined from parallax measurements using equations F4 through F6.
Equation F4  
Equation F5  
Equation F6 
b. Elevations by Parallax
Figure F11 shows the basic parallax and elevation geometry for two points.
Figure F11 Parallax and Elevation Difference 
If we can measure the parallax of both points, we can determine one point's elevation based on the other, Equation F7.
Equation F7 
5. Examples
Example 1
Two vertical control points appear on adjacent aerial photographs. Photography flying height is unknown although it is needed for later computations. The elevation and and x coordinate on each photo for the points are:
Point  Elev (ft)  x (mm)  x' (mm) 
T34  882.53  60.922  35.909 
M60  849.93  29.538  66.379 
Determine the flying height.
Solution
Compute parallax for both points
Rearrange Equation E8 to solve flying height
H = 4.30x10^{3} ft
Example 2
A telecommunication tower appears on overlapping photographs. Coordinates of its top and bottom are:
Left  Right  
Point  x, mm  y, mm  x, mm  y, mm 
Bottom  23.84  15.61  12.41  15.62 
Top  29.75  47.85  18.23  47.89 
The terrain is generally flat and elevation at the tower base 1125 ft. The flying height was 3550 ft and camera focal length 152.44 mm. What is the tower height?
Solution
Compute parallax for tower top and bottom
Use Equation E7 but don't add the tower's base elevation.
Elev = 590 ft