--- title: "Optical distance principle, tangential horizontal distance" subject: "Surveying II" chapter: 2 key: "semester_4/surveying_ii:optical-principle-tangential-horizontal-distance" source: https://www.bctengineer.com/q/semester_4/surveying_ii/optical-principle-tangential-horizontal-distance site: BCT Engineering --- # Optical distance principle, tangential horizontal distance Surveying II · Chapter 2: Tacheometry - Asked in: 2081 Chaitra, 2+4 marks - Video: [Tangential Method of Tacheometric Surveying](https://www.youtube.com/watch?v=Vt0gh_NqghA) — CIVIL Solved - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/optical-principle-tangential-horizontal-distance - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2 ## Question Explain the principle of optical distance measurement. Discuss measurement of horizontal distance by tangential method. ## Short answer Optical distance measurement derives distance from a known base and a measured small angle: D=s/2 cot(β/2). In the tangential method, vertical angles to two targets s apart are measured; D=s/(tanθ1-tanθ2), V=D tanθ2, RL=axis+V-h. For both depression angles, D=s/(tanα1-tanα2) and RL=axis-H tanα1-h. ## Answer **Principle:** in optical distance measurement the distance to a point is found with the instrument alone, from a small angle subtended at the instrument by a known length (or from the length subtended by a known angle), instead of taping the distance. It rests on the geometry of a thin isosceles triangle: $$ D=\frac{s}{2}\cot\frac{\beta}{2}\approx\frac{s}{\beta} $$ where $s$ is the base (known length) and $\beta$ the parallactic angle at the instrument. ```text Instrument Staff / bar O <-------------- D --------------> A \ beta | s \_________________________________ B ``` Depending on which quantity is fixed: 1. **Stadia method:** the angle $\beta$ is fixed by two stadia hairs in the telescope, and the intercept $s$ on the staff varies. $D=Ks+C$. 2. **Subtense method:** the length $s$ is fixed (a subtense bar, 2 m long) and the angle $\beta$ is measured with the theodolite. $D=\frac{s}{2}\cot\frac{\beta}{2}$. 3. **Tangential method:** the vertical angles to two targets a known distance apart on a staff are measured. $D=s/(\tan\theta_1-\tan\theta_2)$. The horizontal distance and the difference in elevation are then found from the distance and the vertical angle, so detail surveys and contouring can be done faster than by chaining, especially in rough ground. #### Horizontal distance by the tangential method In the **tangential method** the stadia hairs are not used. The telescope is pointed at two targets (vanes) on a staff, a known distance $s$ apart, and the two vertical angles are measured. The distance is calculated from the tangents of these angles. It is used when the telescope has no stadia hairs, or when the staff is too far away for the stadia hairs to be read accurately. ```text _ B (upper target) _ - | theta1 - | s O ---------- A (lower target) theta2 \_ | axis ----------\__ | horizontal D ``` Let $D$ be the horizontal distance of the staff from the instrument axis, and $V$ the vertical height of the lower target above (or below) the axis. #### Horizontal distance and vertical distance When both angles are angles of elevation, $\theta_1$ (upper target) and $\theta_2$ (lower target), with the vanes $s$ apart: $$ D=\frac{s}{\tan\theta_1-\tan\theta_2},\qquad V=D\tan\theta_2 $$ $$ \text{RL of staff station}=\text{RL of axis}+V-h $$ where $h$ is the height of the lower vane above the ground at the staff station. When both are angles of depression, $\alpha_1$ (lower target) and $\alpha_2$ (upper target), with $\alpha_1>\alpha_2$: $$ D=\frac{s}{\tan\alpha_1-\tan\alpha_2},\qquad \text{RL}=\text{RL of axis}-D\tan\alpha_2-s-h $$ The method needs only the vertical circle of the theodolite and can be used at long distances, but the angles must be measured carefully, because an angle error $d\theta$ is magnified in $D$ by the small difference $(\tan\theta_1-\tan\theta_2)$ in the denominator. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Optical distance measurement determines the distance from: A. Timing of a pulse over the line B. An angle subtended by a known length at the instrument (thin isosceles triangle) C. Pacing the line D. Weighing a tape under tension Answer: B. A small angle β subtended by a known base s gives D = (s/2)cot(β/2) ≈ s/β. (Key point: Principle of optical distance measurement) 2. In the stadia method: A. The angle is fixed by the hairs and the intercept varies, D = Ks + C B. The length is fixed and the angle is measured C. Two vertical angles are measured to two targets D. No instrument is needed Answer: A. Stadia hairs fix β; the staff intercept s varies with distance. (Key point: Stadia method) 3. In the tangential method with both angles of elevation θ1 (upper) and θ2 (lower) to targets s apart: A. D = s/(tanθ1 + tanθ2) B. D = s/(tanθ1 - tanθ2) C. D = s(tanθ1 - tanθ2) D. D = s cos^2θ1 Answer: B. s = D(tanθ1 - tanθ2), so D = s/(tanθ1 - tanθ2). (Key point: Tangential formula) 4. Which is true of the tangential method? A. It needs stadia hairs B. It uses only the vertical circle and two targets on the staff C. It measures the horizontal angle subtended D. It is more accurate than stadia for short lines Answer: B. No stadia hairs are used; two vertical angles to targets a known distance apart are measured. (Key point: Tangential method features) ## Related in Surveying II - [Principle of optical distance measurement](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aoptical-distance-measurement-principle) — Surveying II, chapter 2 - [Methods of tacheometry, tangential horizontal distance](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atacheometry-methods-tangential-horizontal-distance) — Surveying II, chapter 2 - [Stadia principle, tangential method depression](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Astadia-principle-tangential-depression) — Surveying II, chapter 2 - [Tangential method distance and elevation](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atangential-method-distance-measurement) — Surveying II, chapter 2 - [Tacheometry principle, stadia distance and RL](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atacheometry-principle-stadia-formula) — Surveying II, chapter 2 - [Tangential system expressions, angles of elevation](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atangential-system-elevation-expressions) — Surveying II, chapter 2 ## Related in other subjects - [Principle applied in two-way linear measurement](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-way-linear-measurement-principle) — Surveying I, chapter 1 - [Distance measurement on sloping ground](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Adistance-on-sloping-ground) — Surveying I, chapter 2 - [Correct horizontal distance 630 m, tape 8.9 N](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-630m-8-9n-30-005) — Surveying I, chapter 2 - [Correct horizontal distance, 28.00 m measured](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-fully-support-28-00m) — Surveying I, chapter 2