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Single Point Fixing - Resection

Single Point Fixing - Resection. often interchangeably called three-point problem (special case of simple triangulation.) locates a single point by measuring horizontal angles from it to three visible stations whose positions are known. weaker solution than intersection.

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Single Point Fixing - Resection

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  1. Single Point Fixing - Resection • often interchangeably called three-point problem(special case of simple triangulation.) • locates a single point by measuring horizontal angles from it to three visible stations whose positions are known. • weaker solution than intersection

  2. Single Point Fixing - Resection • extremely useful technique for quickly fixing position where it is best required for setting-out purposes. • theodolite occupies station P, and angles  and  are measured between stations A and B, and B and C.

  3. Single Point Fixing - Resection (Analytical Method) Let BAP = , then BCP = (360° -  -  - ) -  = S -  •  is computed from co-ordinates of A, B and C •  S is known From PAB, PB = BA sin  / sin  (1) From PAB PB = BC sin(S - ) / sin  (2)

  4. Single Point Fixing - Resection (Analytical Method) sin S cot  - cos S = Q  cot  = (Q + cos S) / sin S • knowing  and (S - ), distances and bearings AP, BP and CP are solved Equating (1) and (2)

  5. Single Point Fixing - Resection (Analytical Method) • co-ordinates of P can be solved with the three values. • this method fails if P lies on the circumference of a circle passing through A, B, and C, and has an infinite number of positions.

  6. Resection - Danger Circle  + B + ABC (obtuse) = 180  (sum of opposite angles of cyclic quad.) Accordingly u + v = 180 sin u = sin v, and (sin u / sin v ) = 1; tan v = 0 • at any position along the circumference,the resected station P will have the same angles  and  of the same magnitudes.

  7. Resection - Danger Circle • though the computations will always give the x and y coordinates of the resected station, those co-ordinates will be suspect in all probability. • In choosing resection station, care should be exercised such that it does not lie on the ircumference of the "danger circle".

  8. Ideal Selection of Existing Control Stations • The best position for station P will be 1) inside the  ABC, 2) well outside the circle which passes through A, B and C, 3) closer to the middle control station.

  9. Example: Resection Refer to Figure,  = 41 20’ 35”  = 48 53’ 12” Control points: XA = 5,721.25, YA = 21,802.48 XB = 12,963.71, YB = 27,002.38, XC = 20,350.09, YC = 24,861.22 Calculate the coordinates of P.

  10. Example: Resection Dist. BC =7690.46004 Brg. BC = 106-09-56.8 Dist. AB =8915.8391 Brg. AB = 54-19-21.5  = 180 - ((106-09-56.8)+(54-19-21.5)) = 128-09-24.6 S = (360 -  -  -) = 141-36-48.4 Q = AB sin /BC sin  =1.322286

  11. Example: Resection cot = (Q + cos S) / sin S  = 49 -04-15.5 BP = AB sin /sin  = 10197.4831 BP = BC sin (S - ) / sin  = 10197.4831 (checks)  CBP = 180 - [ + (S - ) ] = 38.5708769° Brg BP = Brg. BC +  CBP = 144 - 44 - 12.0

  12. Example: Resection Ep = EB + BP sin (BRG BP) = 18851.076 Np = NB + BP cos (BRG BP) = 18676.061 • Checks can be made by computing the coordinates of P using the length and bearing of AP and CP.

  13. Intersection • used to increase or densify control stations in a particular survey project • enable high and inaccessible points to be fixed. • the newly-selected point is fixed by throwing in rays from a minimum of two existing control stations • these two (or more) rays intersect at the newly-selected point thus enabling its co-ordinates to be calculated.

  14. Intersection • field work involves the setting up of the theodolite at each existing control station, back-sighting onto another existing station, normally referred to as the reference object (i.e. R.O.), and is then sighted at the point to be established. • normally a number of sets of horizontal angle measurements made with a second-order theodolite (i.e. capable of giving readings to the nearest second of arc) will be required to give a good fix. • intersection formulae for the determination of the x and y co-ordinates of the intersected point may be easily developed from first principle:

  15. Intersection • Let the existing control stations be A(Xa, Ya) and B(Xb,Yb) and from which point P(X, Y) is intersected.  = bearing of ray AP  = bearing of ray BP. • It is assumed that P is always to the right of A and B. ( &  is from 0 to 90)

  16. Intersection Similarly

  17. Intersection Similarly

  18. Intersection

  19. Intersection If the observed angles into P are used, the equation become The above equation are also used in the direct solution of triangulation. Inclusion of additional ray from C, affords a check on the observation and computation.

  20. Where do you want to go ? Global Positioning System Back to Control Survey - Main Menu

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