---
title: "Short note: Relief displacement expression"
subject: "Surveying II"
chapter: 7
key: "semester_4/surveying_ii:relief-displacement-expression"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/relief-displacement-expression
site: BCT Engineering
---

# Short note: Relief displacement expression

Surveying II · Chapter 7: Photogrammetry and Remote Sensing

- Asked in: 2078 Poush; 2078 Baisakh; 2077 Chaitra, 4 marks; 2071 Bhadra, 4 marks; 2076 Bhadra, 4 marks (asked 5 times)
- Video: [What is Relief Displacement in Aerial Photogrammetry? Derivation of Relief Displacement.](https://www.youtube.com/watch?v=bW2RceUZ50c) — Salim K Tutorials
- 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-expression
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-7

## Question

Describe relief displacement and derive its expression in case of vertical photogrammetry.

## Short answer

Relief displacement is the radial image shift on a vertical photo due to height h above datum: d = r h / H, where r is radial distance from principal point and H is flying height. It is derived from similar triangles: r = fR/(H–h), r₀ = fR/H, giving d = r – r₀ = r h / H.

## Answer

**Relief displacement** is the radial shift in the position of the image of a point on a vertical photograph caused by its elevation above or below the datum. Points above the datum are displaced away from the principal point (which is also the nadir point on a vertical photo); points below the datum are displaced towards it. Points lying on the datum are not displaced.

```text
          O  (camera)
         /|\
      f / | \
       /  |  \ H
    a'  a |   \
   -------n-----  photo plane
            \ h  A
 ground   --A0--n'--     datum
```

#### Expression
Let A be the top of an object of height $h$ above the datum, A₀ its base on the datum, $H$ the flying height above the datum, and $R$ the horizontal ground distance from the nadir point to A. Let $a$ and $a_0$ be the images of A and A₀, and $r$ and $r_0$ their radial distances on the photo from the nadir point (the principal point of a vertical photo).

By similar triangles in the camera:

$$
r = \frac{fR}{H-h},\qquad r_0 = \frac{fR}{H}
$$

Relief displacement $d = r - r_0$:

$$
d = fR\left[\frac{1}{H-h}-\frac{1}{H}\right] = \frac{fRh}{H(H-h)} = \frac{r\,h}{H}
$$

since $r = fR/(H-h)$. So

$$
d = \frac{r\,h}{H}
$$

**Conclusions:** the displacement increases with the radial distance $r$ and the height $h$, and decreases as the flying height $H$ increases; it is zero at the principal point.

## Check yourself

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

1. For a point above the datum, relief displacement on a vertical photo is directed:
   A. Towards the principal point
   B. Away from the principal point
   C. Parallel to the flight line
   D. Perpendicular to the radial line
   Answer: B. Points above datum shift radially outward from the nadir/principal point; points below shift inward. (Key point: Definition and direction)

2. Which pair of image distances is used in the derivation?
   A. r = fR/H and r0 = fR/(H+h)
   B. r = fH/R and r0 = fH/(R-h)
   C. r = f(H-h)/R only
   D. r = fR/(H-h) and r0 = fR/H
   Answer: D. Similar triangles give r = fR/(H-h) for the top and r0 = fR/H for the base; d = r - r0. (Key point: Derivation)

3. What is the final expression?
   A. d = rH/h
   B. d = rh/H
   C. d = fh/H
   D. d = h/(rH)
   Answer: B. d = fRh/(H(H-h)) = rh/H since r = fR/(H-h). (Key point: Final expression)

4. What does the expression show about displacement?
   A. It increases with flying height
   B. It is independent of h
   C. It is maximum at the principal point
   D. It is zero at the principal point and decreases with higher H
   Answer: D. d is proportional to r and h and inversely to H, so zero at the principal point (r = 0). (Key point: Dependence)

## Related in Surveying II

- [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×
- [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
- [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
- [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
- [Analytical resection, derive coordinate expression](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanalytical-resection-derive-three-known) — Surveying II, chapter 5
- [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

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