--- title: "Inaccessible object, stations at different levels" subject: "Surveying II" chapter: 3 key: "semester_4/surveying_ii:inaccessible-object-different-levels-same-plane" source: https://www.bctengineer.com/q/semester_4/surveying_ii/inaccessible-object-different-levels-same-plane site: BCT Engineering --- # Inaccessible object, stations at different levels Surveying II · Chapter 3: Trigonometric Leveling - Asked in: 2071 Bhadra, 8 marks - Video: [BASE OF THE OBJECT INACCESSIBLE: INSTRUMENT STATION IN THE SAME VERTICAL PLANE WITH THE ELEVATED OBJ](https://www.youtube.com/watch?v=1rX0_h56_24) — civil solution - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/inaccessible-object-different-levels-same-plane - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-3 ## Question How can you measure the horizontal distance and elevation of an inaccessible object when the instrument positions are at very different levels; instrument stations and the elevated object are in the same vertical plane. ## Short answer Use the single-plane method: measure base AB, vertical angles to P from A and B, and a vane height at B to find axis A. Let x be distance from A to P; solve x from equal RL equations, then RL P = axis A + x tan elevation A. Add curvature-refraction correction for long sights. ## Answer When the base of the object is inaccessible and the two instrument stations A and B are at very different levels but in the same vertical plane as the object P, the **single-plane method** is used. The stations are in line with P, so the horizontal distance and the RL of P are found from vertical angles only, with a measured base AB. #### Field work 1. Set up the theodolite at A (the farther station), level it and measure the height of instrument $h_A$. Measure the vertical angle $\alpha_A$ to P (face left and face right, mean value). 2. Place a vane (height $s$ above the foot) at B and measure the vertical angle $\alpha_{AB}$ from A to the vane. Measure the horizontal distance $D$ = AB. 3. Shift the instrument to B (measure $h_B$) and measure the vertical angle $\alpha_B$ to P. 4. Sight a staff on a BM from the station to obtain the RL of the axis, or use the known RL of B. ```text P _.--'| _.--' | B _.--' aB | A _.--' aA - - - - - - - - - | |<--- D --->|<--- (x - D) --->| |<------------ x ------------>| ``` #### Computation Let $x$ = horizontal distance from A to P, so the distance from B to P is $x - D$. - RL of axis of B $= RL_B + h_B$. - RL of axis of A from the sight A to the vane at B (vertical angle $\alpha_{AB}$, positive if elevation): $$ RL_{\text{axis }A} = RL_B + s - D\tan\alpha_{AB} $$ The height of P is the same from both stations: $$ RL_{\text{axis }A} + x\tan\alpha_A = RL_{\text{axis }B} + (x-D)\tan\alpha_B $$ $$ \boxed{x = \frac{(RL_{\text{axis }B} - RL_{\text{axis }A}) - D\tan\alpha_B}{\tan\alpha_A - \tan\alpha_B}} $$ - Horizontal distance of the object from B: $x - D$ - RL of P: $$ \boxed{RL_P = RL_{\text{axis }A} + x\tan\alpha_A = RL_{\text{axis }B} + (x-D)\tan\alpha_B} $$ If P is a point on top of an object whose base RL is wanted, subtract the height of the object. When the line of sight is long, add the correction for curvature and refraction, $0.0673\,d^2$ (d in km), to each height. Angles read as zenith angles are changed with $\alpha = 90^\circ - z$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which method is used when the base of the object is inaccessible and both stations are in line with it at very different levels? A. Reciprocal levelling with simultaneous angles only B. A single-station method with one vertical angle C. The double-plane (two station) method, using vertical angles and a measured base AB D. Triangulation with horizontal angles at both stations Answer: C. Stations in line with P allow the distance and RL of P to be found from vertical angles and base AB. (Key point: The method) 2. Why is a vane at B sighted from A? A. To measure a horizontal angle B. To get the RL of axis A: RL axis A = RL B + s - D tan(alpha AB) C. To find the horizontal distance AB D. To apply the refraction correction only Answer: B. The sight to the vane ties the axis of A to the known RL of B. (Key point: Purpose of the vane at B) 3. What condition gives the horizontal distance x? A. The two angles of elevation are equal B. The two instrument heights are equal C. The distance A to P equals B to P D. The RL of P found from axis A equals the RL of P found from axis B Answer: D. RL axis A + x tan(aA) = RL axis B + (x - D) tan(aB); solving gives x. (Key point: Condition used to find x) 4. What is done when the lines of sight are long, and how are zenith angles used? A. The correction 0.0673 d^2 (d in km) is added to each height; zenith angles use alpha = 90° - z B. The correction 0.0673 d (d in km) is subtracted; alpha = z C. The correction 6.73 d^2 is added; alpha = z - 90° D. No correction is ever needed Answer: A. Curvature and refraction add about 0.0673 d^2 (d in km) to each height. A zenith angle z gives elevation 90° - z. 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