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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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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
single point fixing resection2
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.
single point fixing resection analytical method
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)

single point fixing resection analytical method4
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)

single point fixing resection analytical method5
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.
resection danger circle
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.
resection danger circle7
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".
ideal selection of existing control stations
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.

example resection
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.

example resection10
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

example resection11
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

example resection12
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.
intersection
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.
intersection14
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:
intersection15
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)
intersection16
Intersection

Similarly

intersection17
Intersection

Similarly

intersection19
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.

where do you want to go
Where do you want to go ?

Global Positioning System

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