---
title: "Relief displacement on a vertical photograph"
subject: "Surveying II"
chapter: 7
key: "semester_4/surveying_ii:relief-displacement-sketch-vertical-photo"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/relief-displacement-sketch-vertical-photo
site: BCT Engineering
---

# Relief displacement on a vertical photograph

Surveying II · Chapter 7: Photogrammetry and Remote Sensing

- Asked in: 2070 Magh, 6 marks
- Video: [Photogrammetry: RELIEF DISPLACEMENT and its DERIVATION](https://www.youtube.com/watch?v=Me9Ubzfu5UY) — Civil Modules
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/relief-displacement-sketch-vertical-photo
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-7

## Question

With the help of neat sketch describe relief displacement on a vertical photograph.

## Short answer

Relief displacement is the outward radial shift of an image point from the photo centre due to elevation; taller objects and points farther from the centre are displaced more. The displacement d = r·h/H, with r radial distance, h object height, H flying height.

## Answer

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.

```text
            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.

## Check yourself

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

1. A tall vertical building appears on a vertical photo as:
   A. A point
   B. A radial line pointing away from the nadir point
   C. A line parallel to flight line
   D. Unchanged
   Answer: B. Top and base images lie on a radial line from n; the top is farther out, causing 'lean'. (Key point: Appearance of vertical object)

2. Base and top image distances are:
   A. r0 = fR/(H-h) and r = fR/H
   B. both equal
   C. r0 = fh/H and r = fh/R
   D. r0 = fR/H and r = fR/(H-h)
   Answer: D. The top is nearer the camera so it has larger r. (Key point: Image distances)

3. d = r - r0 simplifies to:
   A. rH/h
   B. rh/H
   C. fh/R
   D. hH/r
   Answer: B. d = fRh/(H(H-h)) = rh/H. (Key point: Expression)

4. Height of an object from its displacement:
   A. h = dr/H
   B. h = Hr/d
   C. h = d/(rH)
   D. h = dH/r
   Answer: D. Rearranging d = rh/H gives h = dH/r. (Key point: Using it)

## Related in Surveying II

- [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×
- [Field procedure of aerial photogrammetry, relief displacement](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aaerial-photogrammetry-field-procedure-relief-displacement) — Surveying II, chapter 7, asked 2×
- [Expression for scale of vertical photograph](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Avertical-photograph-scale-expression) — Surveying II, chapter 7
- [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

## Related in other subjects

- [Rigid bar AB: vertical displacement of F](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Arigid-bar-ab-steel-aluminium-rods) — Strength of Materials, chapter 3
- [Truss analysis by displacement method](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Atruss-displacement-method-cable-8cm2) — Theory of Structures II, chapter 3
- [Truss: stiffness method, displacement at node A](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Atruss-stiffness-200kn-displacement-a) — Theory of Structures II, chapter 3
- [Jib crane: matrix displacement method](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Ajib-crane-matrix-displacement-10kn) — Theory of Structures II, chapter 3