---
title: "Expression for scale of vertical photograph"
subject: "Surveying II"
chapter: 7
key: "semester_4/surveying_ii:vertical-photograph-scale-expression"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/vertical-photograph-scale-expression
site: BCT Engineering
---

# Expression for scale of vertical photograph

Surveying II · Chapter 7: Photogrammetry and Remote Sensing

- Asked in: 2070 Bhadra, 6 marks
- Video: [Scale of a vertical photograph, Advanced surveying - photogrammetric surveying part 09, Globemech,](https://www.youtube.com/watch?v=l-Uh1HU5WBA) — Globe Mech
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/vertical-photograph-scale-expression
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-7

## Question

Determine the expression for the scale of vertical photograph.

## Short answer

From similar triangles, the scale S of a vertical photograph equals the ratio of photo distance to ground distance: S = f/(H-h), where f is focal length, H is flying height above datum, and h is ground elevation. For level ground at datum, S = f/H.

## Answer

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

```text
        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 $S = 0.152/2500 = 1:16\,447$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The scale expression comes from:
   A. Pythagoras only
   B. Similar triangles Oab and OAB
   C. Sine rule
   D. Parallax
   Answer: B. Triangles Oab and OAB are similar as ab is parallel to AB for a vertical photo. (Key point: Derivation basis)

2. The perpendicular heights of O above photo and ground planes are:
   A. H and h
   B. f and H+h
   C. h and f
   D. f and (H-h)
   Answer: D. Ratio ab/AB = f/(H-h). (Key point: Heights in triangles)

3. For flat ground at datum:
   A. S = H/f
   B. S = f/H
   C. S = f/h
   D. S = fH
   Answer: B. Setting h = 0 gives S = f/H. (Key point: Flat ground)

4. f = 152 mm, H = 3000 m, ground at 500 m gives scale about:
   A. 1:19 737
   B. 1:3000
   C. 1:500
   D. 1:16 447
   Answer: D. S = 0.152/2500 = 1:16 447. (Key point: Numerical example)

## Related in Surveying II

- [Types of aerial photograph, scale relation](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aaerial-photo-types-scale-relation) — Surveying II, chapter 7
- [Relief displacement on a vertical photograph](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Arelief-displacement-sketch-vertical-photo) — Surveying II, chapter 7
- [Short note: Photo scale, photogrammetry uses](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aphoto-scale-photogrammetry-uses) — Surveying II, chapter 7
- [Flying height from photo and map scale](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aflying-height-pq-20cm-focal-24) — Surveying II, chapter 7
- [Short note: Relief displacement expression](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Arelief-displacement-expression) — Surveying II, chapter 7, asked 5×
- [Derive RL on parabolic vertical curve](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aparabolic-vertical-curve-rl-expression) — Surveying II, chapter 6

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