--- title: "Resection by three point problem" subject: "Surveying I" chapter: 6 key: "semester_3/surveying_i:three-point-resection-sketches" source: https://www.bctengineer.com/q/semester_3/surveying_i/three-point-resection-sketches site: BCT Engineering --- # Resection by three point problem Surveying I · Chapter 6: Plane Table Survey - Asked in: 2058 Chaitra, 10 marks - Video: [Three Point Problems || Resection method || Plane table Surveying](https://www.youtube.com/watch?v=AFeqkrvHaIU) — Civil Solution Study - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/surveying_i/three-point-resection-sketches - Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-6 ## Question Explain with the help of neat sketches resection by the three point problem. ## Short answer Resection by the three-point problem locates the plane table station P using plotted points a,b,c. In the tracing-paper method, rays from p to A,B,C on a sheet are shifted until they pass through a,b,c simultaneously; p is pricked through. Lehmann's trial method draws back-rays from a,b,c, forming a triangle of error reduced by Lehmann's rules; the solution fails on the danger circle through A,B,C. ## Answer **Three-point problem:** given three well-defined points A, B and C, plotted on the sheet as **a, b, c**, locate on the sheet the station P where the table is set up and from which A, B and C are all visible. It is a resection method of plane tabling. Three solutions are used: 1. **Mechanical method (tracing paper method)** 2. **Graphical method** (Bessel's method, or the method of perpendiculars) 3. **Trial and error method (Lehmann's method)** #### Mechanical (tracing paper) method 1. Fix a sheet of tracing paper over the drawing sheet. Mark a point **p** on it to represent the station P. 2. Set up the table at P, level it, and orient it roughly with the compass. 3. With the alidade centred on **p**, sight A, B and C and draw rays on the tracing paper. 4. Loosen the tracing paper and move it until the three rays pass through a, b and c at the same time. 5. Prick the point p through onto the drawing sheet. This gives the position of P. 6. Orient the table by placing the alidade along p-a and turning the board until A is bisected. Check by sighting B and C. ```text a b c (on sheet) \ | / \ | / \ | / rays from p (tracing paper) ``` #### Trial and error (Lehmann's) method 1. Set up the table at P, level it and orient it by eye or compass as accurately as possible. 2. Pivoting on a, b and c, sight A, B and C and draw the three rays back. If the orientation is wrong, they form a small **triangle of error**. 3. Estimate the position of P (the point p) inside or near the triangle using the rules below. Re-orient the board by placing the alidade along p-a and turning the table until A is sighted. 4. Repeat until the triangle of error reduces to a point. That point is the position of P. **Lehmann's rules for locating p:** 1. If P lies inside the great triangle ABC, p lies inside the triangle of error; otherwise it lies outside it. 2. When p is outside, it lies on the *same side* of all three rays, that is, to the right of all of them or to the left of all of them, when looking along each ray towards its station. 3. The distance of p from each ray is proportional to the distance of that station from the table. So p is nearest to the ray of the nearest station and farthest from the ray of the farthest station. ```text ray a ray b \ / \____/ /\ /\ <- small triangle / \/ \ of error ray c ``` #### Strong and weak positions - The position is well fixed when P is inside the great triangle ABC, or outside it but well away from the circle through A, B and C. - If P lies on or very near the **circle through A, B and C** (the "danger circle"), the three rays meet in a single point for any orientation of the table, so the problem has no definite solution. Another set of points should then be chosen. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is given and found in the three-point problem? A. Three stations given; an object is found B. Two points given; a base line found C. Three plotted points are given; the table station is found D. Three angles given; a distance found Answer: C. It is a resection method. (Key point: Three-point problem) 2. In the tracing paper method, what is done with the rays? A. Measure the rays with a scale B. Draw rays on the sheet from a, b, c C. Rotate the board until the needle is central D. Move the tracing paper until all three rays pass through a, b and c, then prick p Answer: D. The point where the rays fit a, b, c gives p. (Key point: Tracing paper) 3. What does the trial and error method do? A. Re-orient the board repeatedly until the triangle of error shrinks to a point B. Measure all distances C. Use a single ray only D. Draw a circle through three points Answer: A. The position is refined using Lehmann's rules. (Key point: Trial and error) 4. What happens if P lies on the circle through a, b, c? A. The solution is exact B. The rays meet at many points and no definite solution exists C. The triangle of error doubles D. The method needs no orientation Answer: B. Then other points must be chosen. (Key point: Dangerous circle) ## Related in Surveying I - [Two-point resection in plane tabling](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-point-resection-plane-table) — Surveying I, chapter 6 - [Two point problem; advantages, disadvantages](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-point-problem-adv-disadv) — Surveying I, chapter 6 ## Related in other subjects - [Resection, two point problem, intersection](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aresection-two-point-problem-ab-p) — Surveying II, chapter 5 - [Derive three point analytical resection](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Athree-point-analytical-resection-derivation) — Surveying II, chapter 5, asked 2× - [Intersection vs resection, three point method](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aintersection-vs-resection-three-point) — Surveying II, chapter 5 - [Three point analytical resection, average coordinates of O](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Athree-point-resection-p-w-d) — Surveying II, chapter 5