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Resection, two point problem, intersection

Short answer

Resection determines an unknown station from angles measured at that point; the two-point problem uses two known stations and two auxiliary unknown points. Intersection finds P from angles at known stations. Here AB=972.013 m, bearing AB=269°42'19". By sine rule, AP=1091.700 m, bearing AP=314°34'55", giving P=(3154.440 mE, 8258.295 mN).

What is resection? Explain the two point problem. What is intersection? The coordinates of known stations A (7492 mN, 3932 mE) and station B (7487 mN, 2960 mE). Calculate the coordinate of unknown point P, where the observed horizontal angles taken to P from A is 44°52'36" and to P from B is 75°33'22" respectively.

Resection

Resection is the method of finding the coordinates of an unknown station P by measuring the horizontal angles at P to three (or two) stations of known coordinates. The instrument is set up only at the unknown point.

Two-point problem

The two-point problem is used when only two stations of known coordinates (A and B) are available. Two points alone cannot fix one unknown station by angles at that station, so two unknown stations (C and D, which are intervisible) are set up. At each of them the angles to A and B and to the other unknown station are observed. The angles of the quadrilateral give the angles at A and B (Hansen's method), and the sine rule then gives the distances and the coordinates of both stations.

Intersection

Intersection is the method of finding the coordinates of an unknown point P by measuring the horizontal angles to P from two (or more) stations of known coordinates. The instrument is set up at the known stations, not at P.

Numerical (coordinates of P from the angles at A and B)

The observed angles are at the known stations A and B, so P is found by intersection. Coordinates are (Easting, Northing): A = (3932 E, 7492 N), B = (2960 E, 7487 N). The angles are ∠BAP=44∘52′36′′\angle BAP = 44^\circ52'36'' at A and ∠ABP=75∘33′22′′\angle ABP = 75^\circ33'22'' at B. P is taken on the north side of AB (the south side would give the mirror image).

AB=(2960−3932)2+(7487−7492)2=972.013 m,bearing of AB=269°42′19"AB = \sqrt{(2960-3932)^2 + (7487-7492)^2} = 972.013\ \text{m}, \quad \text{bearing of AB} = 269°42'19"

∠APB=180∘−44∘52′36′′−75∘33′22′′=59°34′02"\angle APB = 180^\circ - 44^\circ52'36'' - 75^\circ33'22'' = 59°34'02"

AP=ABsin⁡75∘33′22′′sin⁡59°34′02"=1091.700 m,BP=ABsin⁡44∘52′36′′sin⁡59°34′02"=795.426 mAP = \frac{AB\sin 75^\circ33'22''}{\sin 59°34'02"} = 1091.700\ \text{m},\qquad BP = \frac{AB\sin 44^\circ52'36''}{\sin 59°34'02"} = 795.426\ \text{m}

Bearing of AP = bearing of AB + 44°52'36" =314°34′55"= 314°34'55" (AB runs almost due west, so turning clockwise puts P to the north).

EP=3932+1091.700sin⁡314°34′55"=3154.440 m,NP=7492+1091.700cos⁡314°34′55"=8258.295 mE_P = 3932 + 1091.700\sin 314°34'55" = 3154.440\ \text{m},\qquad N_P = 7492 + 1091.700\cos 314°34'55" = 8258.295\ \text{m}

Check from B: bearing of BP =89°42′19"−75∘33′22′′=14°08′57"= 89°42'19" - 75^\circ33'22'' = 14°08'57" gives E=3154.440E = 3154.440, N=8258.295N = 8258.295 (same).

Answer: P = (3154.440 E, 8258.295 N).

✓ Answer checked by Claude Opus 5.5 (high effort)

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