1. 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

2. 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

3. 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

4. Quadrants 1st Quadrant : E = +; N = + 2nd Quadrant : E = +; N = -
3rd Quadrant :
E = -; N = -
4th Quadrant :
E = -; N = +

5. 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.

6. Bearing Determination (con’t)
Quadrant
Formula I II
III IV
no change
180 - q +
360
E/N must be calculated ignoring the respective signs of E and N

7. 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

8. 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

9. Example - Join Calculation
In a road scheme, let the coordinates of a point X on the road centreline be mE, mN. This point is to be set out by polar coordinates from a nearby control station Y with coordinates mE, mN.

10. Example - Join Calculation
EYX =
= m
NYX =
= m
distance YX = ( )2 +( )2
= m

11. Example - Join Calculation
 YX = tan-1 ( /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)
= m

12. 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)

13. 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

14. Example - Polar Ray Calculation
If NB = m and EB = m; bearing BA = 25 30’ 41” and distance BA = m, calculate the coordinates of A.
NA = NB + d cos BA
= ( x cos 25 30’ 41”)
= m
EA = EB + d sin BA
= ( x sin 25 30’ 41”)
= m

15. 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)

16. 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)

17. Example: P  R Enter horizontal distance Press P  R
Enter bearing (or azimuth)
Press =
Display  N
Press X  Y
Display  E

18. Example: R  P Enter  N Press R P Enter  E Press =
Display horizontal distance
Press X  Y
Display angle

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