---
title: "Anallatic tacheometer A to B distance, gradient"
subject: "Surveying II"
chapter: 2
key: "semester_4/surveying_ii:anallatic-bm-b-distance-gradient-970"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/anallatic-bm-b-distance-gradient-970
site: BCT Engineering
---

# Anallatic tacheometer A to B distance, gradient

Surveying II · Chapter 2: Tacheometry

- Asked in: 2069 Bhadra, 10 marks
- Video: [Determination of gradient between two points in tacheometric surveying- part 1](https://www.youtube.com/watch?v=FJos96jcQPQ) — Gayathri VG
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/anallatic-bm-b-distance-gradient-970
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

The following readings were taken by a tacheometer with the staff held vertical. The tacheometer is fitted with an anallatic lens and the multiplying constant is 100. Find out the horizontal distance from A to B and gradient of AB.

| Instrument station | Staff Station | Vertical angle | Staff readings | Remarks |
|---|---|---|---|---|
| A | BM | -6°30' | 1.100, 1.153, 2.060 | RL of BM = 970.00 m |
| | B | +10°0' | 0.982, 1.105, 1.188 | |

## Short answer

For BM (s=0.960 m, −6°30′) and B (s=0.206 m, +10°00′), compute D=Ks cos²θ and V=½Ks sin2θ with K=100, C=0. RL of instrument axis from BM is 981.951 m. Horizontal distance AB = 19.979 m. Gradient from instrument at A to B is rising 1 in 8.3 (12.10%).

## Answer

Anallatic lens: $C=0$, $K=100$, staff vertical.

**Reading of the data:** no bearings or height of instrument are given, so the horizontal distance from A to B is the distance from the instrument at A to the staff at B. The middle hair readings are used as printed (1.153 m for BM and 1.105 m for B). Since the height of instrument is not given, the gradient is taken from the instrument axis at A (RL of axis) to the staff point B.

$$
D=Ks\cos^2\theta,\qquad V=\tfrac12Ks\sin2\theta
$$

#### Step 1: Sight on the benchmark (angle of depression $6^\circ30'$)
**Point BM** (angle of depression $6^\circ 30'$):

$$
\begin{aligned}
s&=2.060-1.100=0.960\ \text{m}\\
D&=100\times0.960\times\cos^2 6^\circ 30'=94.770\ \text{m}\\
V&=\tfrac12\times100\times0.960\times\sin 2(6^\circ 30')=-10.798\ \text{m}
\end{aligned}
$$

#### Step 2: RL of the instrument axis
$$
\text{RL}_{BM}=\text{RL}_{axis}+V-h\ \Rightarrow\ \text{RL}_{axis}=970.000-(-10.798)+1.153=981.951\ \text{m}
$$

#### Step 3: Sight on B (angle of elevation $10^\circ$)
**Point B** (angle of elevation $10^\circ 00'$):

$$
\begin{aligned}
s&=1.188-0.982=0.206\ \text{m}\\
D&=100\times0.206\times\cos^2 10^\circ 00'=19.979\ \text{m}\\
V&=\tfrac12\times100\times0.206\times\sin 2(10^\circ 00')=3.523\ \text{m}
\end{aligned}
$$

$$
\text{RL}_B=981.951+3.523-1.105=984.369\ \text{m}
$$

#### Step 4: Distance and gradient
Horizontal distance from A to B:

$$
D_{AB}=19.979\ \text{m}
$$

$$
\text{Gradient}=\frac{\text{RL}_B-\text{RL}_{axis}}{D_{AB}}=\frac{+2.418}{19.979}=0.12102
$$

**Answer:** horizontal distance AB = **19.979 m**; RL of B = 984.369 m; gradient rising **1 in 8.3 (12.10 %)** from the instrument level at A to B.

Note: the middle hair reading for BM (1.153) is not the mean of the outer hairs (1.580). If 1.580 m were the correct reading, the RL of the axis would be 982.378 m and the RL of B 984.796 m, but the distance and the gradient above would be unchanged.

## Check yourself

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

1. Why is the staff first sighted on the BM?
   A. To find the RL of the instrument axis
   B. To find the distance AB
   C. To find the bearing of AB
   D. To set K
   Answer: A. RL_BM = RL_axis + V - h, so RL_axis = 970 - V + h. (Key point: Purpose of the BM sight)

2. With V = -10.798 and h = 1.153, what is RL of axis?
   A. 958.049 m
   B. 981.951 m
   C. 970.000 m
   D. 959.045 m
   Answer: B. RL_axis = 970 + 10.798 + 1.153 = 981.951 m. (Key point: RL of axis)

3. What is the horizontal distance from A to B?
   A. 94.770 m
   B. 114.749 m
   C. D = Ks cos^2θ at B = 19.979 m
   D. 3.523 m
   Answer: C. It is the distance from the instrument at A to the staff at B, using the sight on B. (Key point: Horizontal distance AB)

4. How is the gradient obtained?
   A. (RL_B - 970)/D
   B. V/h
   C. falling 1 in 8.3
   D. (RL_B - RL_axis)/D_AB = +2.418/19.979, rising about 1 in 8.3
   Answer: D. No HI is given, so gradient is measured from the axis at A to B. (Key point: Gradient)

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