Influence of Tilt on Geometry of the Conventional Aerial Photograph: Types of Tilts, Effect on Geometry and Scale (Especially for GATE-Geospatial 2022)

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It is obvious from the earlier discussion on flying for aerial photography that it is impossible to obtain tilt-free photographs.

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Generalised Coordinates of a Rigid Body Showing Three Types of Tilting About the Three Coordinate Axes

  • The movements of the aircraft with reference to the three mutually perpendicular X-, Y- and Z-axes give rise to the three major types of tilt:
Generalised Coordinates of a Rigid
  • (Phi) -tilt or longitudinal tilt or tip is the first or primary rotation taken about the Y-axis;
  • (omega) -tilt or lateral tilt is the second or the secondary rotation taken about the X-axis; and
  • K (Kappa) or swing is the third or tertiary rotation taken about the Z-axis. It should be noted that the rotation angles shown in Figure below are measured in a clockwise direction and indicate positive direction, the so-called right-handed system adopted for a negative.

The Image-Ground Relationship for a Tilted Photograph

With tilting, the geometrical relationship between the camera and the ground will be changed as shown in Figure below. The tilt is, in fact, the angle formed between the optical axis and the vertical line (i.e.. t in Figure below) , or the angle which the plane of the photograph makes with the horizontal plane.

The Image-Ground Relationship for a Tilted Photograph
  • Clearly, with the presence of tilts, the optical axis of the camera is no longer pointing vertically, and where the vertical line cuts the plane of the photograph a nadir point (n) is found. It is the point on the photograph vertically beneath the camera. A corresponding ground point N also exists. The optical axis and the vertical line together constitute a vertical plane (nLp or NLP) known as the principal plane.
  • The intersection of the principal plane with the plane of the photograph gives a line called the principal line of tilt. The direction of the principal plane of the photograph on the ground can be obtained if the north point is known and its bearing is the clockwise angle measured horizontally about the ground nadir point, N, as angle ap. This is, in fact, the ground direction of the tilt since the angle of tilt lies in the principal plane. The angle of tilt can be bisected and the angle bisector meets the photograph along the principal line at a point called the isocentre (i) with a corresponding point on the ground at I.
  • The point i is at a distance of f. tan (t/2) from the principal point where f is the focal length, with reference to the right-angled triangle Lnp in Figure above. From the isocentre, a perpendicular line can be drawn which indicates the axis of tilt. This horizontal line which passes through the isocenter is known as the isometric parallel because the tilting has no effect on its axis and hence the scale along it is not changed at all. Other lines that can be drawn parallel to this axis of tilt are known as plate parallels.
  • The graphics and metric effects of the tilts on the resulting photograph may best be seen by assuming that the ground is flat and is covered by a square net. The truly vertical aerial photograph will give a correct representation of this at a reduced scale as shown in below Figure-a. But if the aerial photograph is tilted, the representation of the shape of the terrain will be distorted as shown in below Figure-b.
  • It is found that the scale changes regularly along the principal line of tilt so that at one end the scale is uniformly too small whilst at the other, the scale is uniformly too large. But on the axis of tilt, no scale changes occur (hence the name isometric parallel) . All displacements resulting are either inwards or outwards from the isocentre depending on whether it is below or above the axis of tilt.

Tilted Photograph Superimposed on Truly Vertical Photograph

The Fiat Terrain by a Truly Vertical Aerial Photograph

Figure- a: Correct representation of the fiat terrain by a truly vertical aerial photograph

Tilted Photograph Superimposed on the Truly Vertical Photogr …

Figure- b: Tilted photograph superimposed on the truly vertical photograph

Geometric Analysis of a Tilted Photograph: Scale of Tilted Photograph

With reference to below Figure, one can mathematically determine the scale of the tilted photograph at the nadir point (n) , principal point (p) , isocentre (i) and any point (m or q) on the plate parallel by the following formulae (which can be easily verified by employing simple trigonometric relationships) :

and (4) scale at m on a plate parallel (i.e.. on the downward side of the isocentre)

the scale at q on a plate parallel (i.e.. on the upward side of the isocentre)

Geometric Analysis of a Tilted Photograph

Geometric analysis of a tilted photograph

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