Join Calculation

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Join Calculation

• calculation of the
• 1) whole circle bearing (or azimuth)

2) distance
between two points (or stations) if the
coordinates of them are known on a grid system
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Join Calculation

• N North direction
• Sta. A and Sta. B stations A and B
• ?AB Bearing AB,
• Azimuth AB or
• WCB AB
• WCB Whole circle bearing

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Procedures

• draw a sketch showing the relative positions of
  the two stations to determine in which quadrant
  the line falls
• the greatest source of error in this type of
  calculation is wrong identification of quadrant

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Quadrants

• 1st Quadrant
• ?E ?N
• 2nd Quadrant
• ?E ?N -
• 3rd Quadrant
• ?E - ?N -
• 4th Quadrant
• ?E - ?N

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Bearing Determination

• ? AB tan -1 (?EAB/?NAB)
• tan -1 (EB - EA) / (NB - NA)
• final value of ? AB will depend on
• the quadrant of the line and
• a set of rules, based on the quadrant in which
  the line falls.

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Bearing Determination (cont)
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Distance Determination

• LAB ??E2 ?N2
• To check the result against gross error use
• LAB (?EAB/sin ?AB) (?NAB/ cos ?AB)
• small differences occur between the two results,
  the correct answer is given by the
  trigonometrical functions

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Bearing Determination

• if ? 5?, L found from (?N/ cos ?) gives the
  more accurate answer than (?E/ sin ?) since the
  cosine function is changing less rapidly than the
  sine function at this angle value
• inspection of the different columns in the
  trigonometrical values for the two functions will
  show which is the slower changing

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Example - Join Calculation

• In a road scheme, let the coordinates of a point
  X on the road centreline be 8 612 910.74 mE, 8
  157 062.28mN. This point is to be set out by
  polar coordinates from a nearby control station Y
  with coordinates 8 613 112.33mE, 8 157 238.91mN.

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Example - Join Calculation
?EYX 8 612 910.74 - 8 613 112.33
-201.59 m ?NYX 8 157 062.28 - 8 157 238.91
-176.63 m distance YX ?(-201.59)2
(-176.63)2
268.02 m
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Example - Join Calculation

• ? YX tan-1 (201.59 /176.63) 48? 46 32
• Since ? YX is in the 3rd quadrant, therefore
• bearing of YX 180? 48? 46 32
• 228? 46 32
• To avoid gross error, check distanceYX using the
  following formulae
• LAB (?EAB/sin ?AB) (?NAB/ cos ?AB)
• 268.02 m

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Polar Ray Calculation

• Name given to the process of determining
  coordinates of one point (EA and NA) based on the
  following known information
• coordinates of another point (EB and NB),
• the bearing ?bA, and
• the distance BA (dBA)

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Polar Ray Calculation
The formulae are as follows NA NB dBA cos
?BA and EA EB dBA sin ?BA

• all additions being algebraic. The result can be
  checked by doing a join calculation

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Example - Polar Ray Calculation
If NB 1068.263 m and EB 2135.920 m bearing
?BA 25? 30 41 and distance BA 100.023m,
calculate the coordinates of A.

• NA NB d cos ?BA
• 1068.263 (100.023 x cos 25? 30 41)
• 1158.534 m
• EA EB d sin ?BA
• 2135.920 (100.023 x sin 25? 30 41)
• 2178.999 m

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Coordinates Computations using Electronic
Calculators

• useful for computing coordinates because the sine
  and cosine of the bearing need not be entered
• coordinate difference of ?E and ?N or bearing
  and distance are then displayed at the press of
  several keys (normally less than the conventional
  keystrokes)

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Coordinates Computations using Electronic
Calculators

• built-in functions P?R and R?P
• P?R is the conversion of polar coordinate into
  rectangular coordinates (Polar Ray Calculation)
• R?P is the reverse conversion (Join Calculation)

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Example P ? R

• Enter horizontal distance
• Press P ? R
• Enter bearing (or azimuth)
• Press
• Display ? N
• Press X ? Y
• Display ? E

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Example R ? P

• Enter ? N
• Press R ?P
• Enter ? E
• Press
• Display horizontal distance
• Press X ? Y
• Display angle

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