--- title: "Reciprocal leveling, RL of hill top relation" subject: "Surveying II" chapter: 3 key: "semester_4/surveying_ii:reciprocal-leveling-when-hill-top-relation" source: https://www.bctengineer.com/q/semester_4/surveying_ii/reciprocal-leveling-when-hill-top-relation site: BCT Engineering --- # Reciprocal leveling, RL of hill top relation Surveying II · Chapter 3: Trigonometric Leveling - Asked in: 2070 Magh, 6 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/reciprocal-leveling-when-hill-top-relation - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-3 ## Question In what situation reciprocal trigonometrically levelling is conducted? Derive a relation for determining the reduced level of a hill top when two instrument stations and target point are at different vertical plane. ## Short answer Reciprocal trig levelling is used when stations are far apart or separated by a river or obstacle and curvature-refraction errors must be eliminated. If P is not in the vertical plane of X-Y, measure base XY and horizontal angles at X and Y; compute XP and YP by sine rule, then RL P = axis RL + horizontal distance times tangent of vertical angle. ## Answer #### Situations for reciprocal trigonometric levelling Reciprocal trigonometric levelling is conducted when: - two stations are far apart or are separated by a wide obstacle such as a river, valley or lake, so that ordinary (spirit) levelling between them is not practical, and - the effects of **curvature and refraction** are large and must be eliminated. Vertical angles are measured from both stations at the same time, using the same type of instrument and the same height of signal, so that the errors of curvature and refraction are equal and opposite and cancel in the mean: $$ h = D\tan\frac{\alpha+\beta}{2} $$ where $\alpha$ is the angle of elevation from the lower station and $\beta$ is the angle of depression from the upper station. #### Hill top when the stations and the target are in different vertical planes If the instrument stations X and Y are not in the same vertical plane as the object P, a base line XY is measured and horizontal angles are observed at X and Y. The horizontal distances XP and YP are found by the sine rule in the plan triangle XYP, and the height of P is found from the vertical angles. ```text P (top) / \ XP / \ YP / \ / \ X -------- Y alpha b beta ``` #### Derivation Let the base line be $b = XY$, the horizontal angles $\alpha = \angle YXP$ and $\beta = \angle XYP$, so $\angle XPY = 180^\circ - (\alpha + \beta)$. $$ \begin{aligned} XP &= \frac{b\sin\beta}{\sin(\alpha+\beta)} \\ YP &= \frac{b\sin\alpha}{\sin(\alpha+\beta)} \end{aligned} $$ Let $\theta_X$ and $\theta_Y$ be the vertical angles (elevation) of P from X and Y, and $h_X$, $h_Y$ the heights of the instrument axes above their stations. If a BM of known RL is sighted from X and Y with back-sights $BS_X$, $BS_Y$: $$ \begin{aligned} RL_{\text{axis X}} &= RL_{BM} + BS_X, \qquad RL_{\text{axis Y}} = RL_{BM} + BS_Y \\ RL_P &= RL_{\text{axis X}} + XP\tan\theta_X \quad (\text{from X}) \\ RL_P &= RL_{\text{axis Y}} + YP\tan\theta_Y \quad (\text{from Y}) \end{aligned} $$ For zenith angle $z$, $\theta = 90^\circ - z$. The mean of the two values is the RL of P. If the stations are at different levels, the difference of the axis RLs is already included through the BM sights. When the sights are long, the correction for curvature and refraction $0.0673\,d^2$ (d in km) is added to each height. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In which situation is reciprocal trigonometric levelling conducted? A. On short sights over flat ground B. When stations are far apart or separated by a wide obstacle, and curvature and refraction are large C. When the stations and the target are close together D. When no theodolite is available Answer: B. Reciprocal observations are used across rivers or valleys, where ordinary levelling fails and curvature and refraction matter. (Key point: When it is conducted) 2. What does measuring vertical angles from both stations at the same time, with the same signal height, achieve? A. It doubles the accuracy of the horizontal angles B. It removes the need for a base line C. Curvature and refraction errors are equal and opposite, so they cancel in the mean D. It removes the instrument height Answer: C. With equal errors at both ends, the mean (alpha + beta)/2 is free of curvature and refraction. (Key point: Why observe from both ends) 3. Triangle XYP has base b = XY, horizontal angle alpha at X and beta at Y. Which gives XP? A. b sin(alpha) / sin(alpha + beta) B. b sin(alpha + beta) / sin(beta) C. b cos(beta) / cos(alpha + beta) D. b sin(beta) / sin(alpha + beta) Answer: D. XP is opposite the angle at Y (beta); the angle at P is 180° - (alpha + beta), so XP = b sin(beta) / sin(alpha + beta). (Key point: Sine rule in the plan) 4. How is the RL of the hill top P finally obtained? A. RL axis X + XP tan(theta X) and RL axis Y + YP tan(theta Y); the mean is taken B. RL of BM + XP C. The sum of the two values from X and Y D. RL of BM + base length b Answer: A. Each station gives a value of RL of P from its own axis RL, distance and elevation. Their mean is the RL of P. (Key point: RL of P) ## Related in Surveying II - [RL of hill top from two stations (A-B 118 m)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-rl-two-stations-2079j) — Surveying II, chapter 3, asked 4× - [RL of hill top Q from stations P and R](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-q-from-p-and) — Surveying II, chapter 3, asked 4× - [RL of hill station P, A-B 150 m, target 3.5 m](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-station-p-vane-15-ab) — Surveying II, chapter 3, asked 2× - [RL of hill station P from stations R and Q](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-station-signal-r-q-p) — Surveying II, chapter 3 - [RL of hill top P (A to B 112 m, RL 1290)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-rl-2077c-ab-112) — Surveying II, chapter 3, asked 4× - [RL of hill top P (A to B 112 m, RL 1280)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-rl-2073m-ab-112) — Surveying II, chapter 3, asked 4× ## Related in other subjects - [Reciprocal levelling; RL of closed traverse stations](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-closed-traverse-bcdefgb) — Surveying I, chapter 5 - [Reciprocal levelling: RL of A, B at 1160.18 m](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-rl-b-1160-18) — Surveying I, chapter 5, asked 2× - [Reciprocal levelling: RL of A, B at 1260.18 m](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-rl-b-1260-18) — Surveying I, chapter 5, asked 2× - [Reciprocal levelling AB 250 m, RL 1450.500](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Areciprocal-leveling-ab-250m-1-252) — Surveying I, chapter 5