Relief displacement on a vertical photograph

Relief displacement is the radial displacement of the image of a point on a vertical photograph due to its elevation above or below the datum plane. On a vertical photograph the displacement is directed outward from the principal point (for high points) or inward towards it (for low points); on a vertical photo the principal point is also the nadir point.

            O (lens)
          / | \
         /  |  \          H = flying height above datum
        /   |   \
   a0  a    n               photo (r0, r)
   ----+----+---
                \   h
      A0     A    N0  (object top A above base A0)
   ============ datum

Explanation with the sketch

• Let a vertical object AA₀ stand on the datum, with its top A at height $h$. Its base A₀ and top A lie on the same vertical line, at horizontal distance $R$ from the ground nadir point N. The vertical plane through this line and the lens O also contains the plumb line ON, so the object images along a radial line through the photo nadir $n$.
• On the photo, the base A₀ forms image $a_0$ at radial distance $r_0 = fR/H$ from the nadir point $n$, and the top A forms image $a$ at the larger distance $r = fR/(H-h)$.
• The vertical object therefore appears as a radial line $a_0a$ pointing away from $n$: this is the "lean" of tall buildings, towers and trees on aerial photographs.

Expression

$$d = r - r_0 = \frac{fRh}{H(H-h)} = \frac{r\,h}{H}$$

• $d \propto r$: no displacement at the centre, maximum at the corners.
• $d \propto h$: taller objects are displaced more.
• $d \propto 1/H$: flying higher reduces displacement.

Uses

Height of an object: $h = \dfrac{dH}{r}$. Relief displacement also creates parallax for stereoscopic viewing, and it has to be removed by rectification to make orthophotos.