Single Point Fixing - Resection

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# Single Point Fixing - Resection - PowerPoint PPT Presentation

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
• 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 - 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)

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 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 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

 + 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 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
• 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

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: 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: 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: 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
• 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.
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:
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)
Intersection

Similarly

Intersection

Similarly

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.

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