--- title: "Reciprocal trigonometrical levelling formula" subject: "Surveying II" chapter: 3 key: "semester_4/surveying_ii:reciprocal-trig-leveling-formula" source: https://www.bctengineer.com/q/semester_4/surveying_ii/reciprocal-trig-leveling-formula site: BCT Engineering --- # Reciprocal trigonometrical levelling formula Surveying II · Chapter 3: Trigonometric Leveling - Asked in: 2075 Baisakh, 4 marks - Video: [Reciprocal Levelling|Curvature, Refraction and Collimation Correction](https://www.youtube.com/watch?v=UYZKNd7XKH0) — Master Civil Engineering - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/reciprocal-trig-leveling-formula - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-3 ## Question Explain about the reciprocal trigonometrical levelling and express the formula for computing elevation difference. ## Short answer Reciprocal trigonometric levelling eliminates curvature and refraction by measuring vertical angles from both ends of a line simultaneously. The elevation difference h is computed as h = D tan((α+β)/2), where D is horizontal distance, α the elevation angle from one station and β the depression angle from the other. ## Answer #### Reciprocal trigonometric levelling When two stations A and B are far apart, or are on opposite sides of a wide obstacle (river, valley), the effects of curvature and refraction cannot be neglected. The vertical angles are then measured from **both** stations, at nearly the same time, so that these errors cancel. From A the angle of elevation $\alpha$ to B is measured, and from B the angle of depression $\beta$ to A is measured, with the signal at each station at the same height as the instrument axis (otherwise the angles are corrected for the difference). ```text B ---- horizontal at B _.-' | (beta = depression to A) _.-' | _.-' | h _.-' | A _.-' alpha | --------------------------------C (level line through A) |<-------------- D ------------>| ``` #### Expression for elevation difference Let $D$ = horizontal distance AB, $\phi = D/R$ the angle between the horizontals at A and B (R = radius of the earth), and $\varepsilon$ the angular effect of refraction, which is assumed equal at both ends because the sights are taken at the same time. If $\theta$ is the true angle of elevation of the chord AB at A, the true angle of depression of the chord at B is $\theta + \phi$, because the horizontal at B is turned through $\phi$ relative to that at A. Refraction raises the apparent position of the target at both ends, so the observed angles are $$ \alpha = \theta + \varepsilon, \qquad \beta = (\theta + \phi) - \varepsilon $$ Adding, refraction cancels: $$ \alpha + \beta = 2\theta + \phi \quad\Rightarrow\quad \theta + \frac{\phi}{2} = \frac{\alpha+\beta}{2} $$ The difference in elevation $h$ is measured along the vertical at B, from the level line through A (point C). The chord AC lies $\phi/2$ below the horizontal at A, so $\angle BAC = \theta + \phi/2$ and $$ \boxed{h \approx D\tan\left(\theta + \frac{\phi}{2}\right) = D\tan\frac{\alpha + \beta}{2}} $$ Both refraction (through $\varepsilon$) and curvature (through $\phi$) drop out, so no separate correction for curvature and refraction is needed. If the signal heights are not equal to the instrument heights, the observed angles are first corrected for the difference (or the difference in heights is allowed for in $h$). Then $RL_B = RL_A + h$. The principal advantage of the method is that the uncertain effects of curvature and refraction are eliminated by taking the mean of the two reciprocal observations. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. When is reciprocal trigonometric levelling used? A. When the two stations are close and the ground is flat B. When stations are far apart or across a wide obstacle and curvature and refraction cannot be neglected C. When only one instrument station is available D. When the staff can be read directly from a level Answer: B. It is used over rivers, valleys or long lines where curvature and refraction are large and must be eliminated. (Key point: When it is used) 2. Why are the vertical angles from A and B measured at nearly the same time? A. So that the instrument heights cancel B. So that the horizontal distance need not be measured C. So that refraction is the same at both ends and cancels in the mean of the angles D. So that curvature is doubled Answer: C. Equal refraction at both ends appears with opposite sign in alpha and beta, so it cancels when they are added. (Key point: Why simultaneous observations) 3. With alpha = theta + epsilon and beta = theta + phi - epsilon, what is the true inclination theta of the chord? A. theta = (alpha + beta)/2 - phi/2 B. theta = (alpha + beta)/2 + phi/2 C. theta = alpha + beta - phi D. theta = (alpha - beta)/2 Answer: A. Adding gives alpha + beta = 2 theta + phi, so theta = (alpha + beta)/2 - phi/2. (Key point: Derivation step) 4. What does the formula for the elevation difference become for short lines? A. h = D tan(alpha - beta) B. h = 2D tan(alpha + beta) C. h = D tan((alpha + beta)/2), since D/2R is negligible D. h = D (alpha + beta) Answer: C. From h = D tan((alpha + beta)/2 - D/2R), the D/2R term is dropped for short lines. 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