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

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

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

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

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.

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

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

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

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.

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

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

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)

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

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

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)

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)

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

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

Where do you want to go? Traversing Back to Traverse - Main Menu