Geometry of an Overlapping Pair of Aerial Photographs and Parallax (Especially for GATE-Geospatial 2022)

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Geometry of an Overlapping Pair of Aerial Photographs

In photogrammetry and photo-interpretation, overlapping pairs of vertical aerial photographs capable of generating three-dimensional views of the area concerned (i.e.. stereo models) are usually employed to allow the extraction of height data and a better appreciation of the terrain.


  • The perception of distance by means of binocular vision depends on the magnitude of the parallactic angle (a) formed by the two eyes at the point of convergence. As shown in Fig below, the convergence of the axes of the eyes at a gives rise to an angle al at a distance D1 and convergence at b gives rise to at .
  • The angular difference, i.e.. () , tells the mind that the distance between a and b is e. By analogy, for an overlapping pair of aerial photographs, the two exposure positions of the aerial camera represent the two big of a giant looking down on to the ground, and by means of the differences in parallactic angles, the heights of various objects from the ground can be determined.
Paralactio Angles and Distances
  • Thus, by having a pair of overlapping photographs taken at two different viewpoints, parallax is produced. Parallax is, therefore, the apparent displacement of the position of a body, with respect to a reference point or system, caused by a shift in the point of observation.
  • There are two types of parallax, the X-parallax and the Y-parallax, according to the direction in which these parallaxes occur during the stereoscopic viewing of the pair of photographs.
  • Y-Parallax, or Want of Correspondence, or vertical parallax, occurs as a result of
    • Unequal flying height,
    • Tilt in the Y-direction and
    • Misalignment of the flight line.
  • For (1) , when the flying heights for the pair of aerial photographs are different (but without any tilts) , the sectional view of the photo-pair becomes as shown in Fig below,
Y Parallax Caused by Unequal Flying Heights
  • An exaggerated case which shows that one photograph is higher than the other, thus resulting in the scale of photo 1 being smaller than that of photo 2. The plan view is shown in Fig below in which the Y-coordinates of (i.e.. yb) is smaller than the Y-coordinate of (i.e.. yb) by an amount (i.e. yib-yb) . The discrepancy of is the Y-parallax of the point.
  • The same kind of discrepancy measured in the Y-axis of the photograph can occur for (2) tilting of one photograph relative to the other in the Y-direction
1-Parallax Caused by the Tilting of Photo

(a) 1-parallax caused by the tilting of photo (2) relative to photo (1) in the 1-direction (b) Plan view of the resultant 1-parallax: since only photo (2) is tilted relative to photo (1) in the 1-direction, a nadir point (n2) can be located on photo (2) as shown.

and for (3) misalignment of the flight line between two photographs.

Y-Parallax Caused by Misalignment of Flight Lines
  • It is obvious that the three effects of (1) , (2) and (3) can be combined to produce even more complex 1-parallax, and can cause great difficulty in stereoscopic viewing of the pair of photographs.
  • X-Parallax is the apparent image displacement created along the X-direction on the stereo pair of aerial photographs.
  • To derive a geometrical definition, one may consider the parallax of a point A in Fig below, which can be defined as the angle subtended at A by the projection centre but also as linear distances measured on the photograph.
Definition of X-Parallax

Thus, with reference to the above Figure, for the negative plane, the parallactic angle () becomes:

For the positive plane, the parallactic angle similarly () becomes:

Developed by: