Expression for scale of vertical photograph

The scale of a photograph is the ratio of a length measured on the photograph to the corresponding length on the ground.

        O  (exposure station)
       /|\
    f / | \
     /  |  \
   a----n----b    photo plane
        |
   H-h  |
        |
   A----N----B    ground at height h
   -----------    datum

Derivation

Let O be the lens centre, $f$ the focal length, $H$ the flying height above datum, and A and B two ground points at elevation $h$ above the datum, whose images are $a$ and $b$ on a vertical photo. The line $ab$ on the photo is parallel to $AB$ on the ground, as the camera axis is vertical.

Triangles $Oab$ and $OAB$ are similar. The perpendicular height of O above the plane of $ab$ is $f$ and above the ground plane AB is $(H-h)$. Hence

$$\frac{ab}{AB} = \frac{f}{H-h}$$ $$\boxed{S = \frac{f}{H-h}}$$

Notes

• For flat ground at the datum ($h = 0$): $S = f/H$.
• If the ground is not level, the scale is different for each elevation; the average scale uses the mean ground elevation $h_{av}$: $S_{av} = f/(H-h_{av})$.
• Scale can also be found from the map: $S = \dfrac{\text{photo distance}}{\text{map distance}}\times\text{map scale}$.
• Example: $f = 152$ mm, $H = 3000$ m above a ground at 500 m gives $S = 0.152/2500 = 1:16\,447$.