4. Curves and Coordinates

a. Coordinate Equations

Equations C-16 and C-17 are general equations for computing coordinates using direction and distance from a known point, Figure C-26.

           Equation C-16 
    Equation C-17 


Figure C-26. Coordinate Computation


Direction (Dir) may be either a bearing or azimuth.

Curve point coordinates can be computed using these equations from a base point. Since the radial chord method uses the BC as one end of all the chords, it can also be used as the base point for coordinate computations.

b. Computation Process

Assuming we start with the tangents and PI, then fit a curve, the general process is as follows:


Figure C-27

      The original tangent lines have directions; PC has coordinates.


Figure C-28


A curve is fit to the tangents.

End points are at distance T from the PI along the tangents.


Figure C-29


Compute coordinates of BC using back-direction of the tangent BC-PI and T.

horiz97       Equation C-18
horiz98   Equation C-19 


Figure C-30


Compute coordinates of EC using direction of the tangent PI-EC and T.

These will be use for a later math check.

horiz99       Equation C-20
 horiz100   Equation C-21


Figure C-31


Use a curve point's deflection angle to compute the direction if its radial chord from the BC.

horiz101       Equation C-22 


δ is positive for right deflections, negative for left.
Using the direction and chord length, compute the point's coordinates.

horiz102       Equation C-23 
 horiz103   Equation C-24 


c. Example

Continuing with the previous example problem.

Summary of given and computed curve data: 

Δ = 55°00'00" R = 500.00 ft  
D = 11°27'33.0" L = 479.965 ft T = 260.284 ft
LC = 461.749 ft E = 63.691 ft M = 56.494 ft


Point Station
PI 25+00.00
BC 22+39.716
EC 27+19.681 Bk = 27+60.284 Ah




Additional information: Azimuth of the initial tangent is 75°40'10"; coordinates of the PI are 1000.00 N, 5000.00' E.

Compute coordinates of the BC:



Compute the coordinates of the EC:



Set up Equatons C-21 through C-24 for this curve.


This is the Radial Chord table computed previously:

Curve Point Arc dist, li, (ft) Defl angle,δi Radial chord, c
EC      27+19.681 Bk 479.965 27°30'00.0" 461.748
       27+00 460.284 26°22'20.4" 444.203
       26+00 360.284 20°38'33.9" 352.540
       25+00 260.284 14°54'47.4" 257.355
       24+00 160.284 9°11'01.0" 159.599
       23+00 60.284 3°27'14.5" 60.248
BC      22+39.716 0.000 0°00'00.0" 0.000


 Add three more columns for direction and coordinates:

     Curve Point Azimuth, Azi North, Ni East, Ei
EC      27+19.681 Bk      
BC      22+39.716      


Complete the table using the three equations for this curve

At 22+39.716, we're still at the BC so the coordinates don't change.

At 23+00:


At 24+00:


and so on for the rest of the curve points.

The completed curve table is: 

     Curve Point Azimuth, Azi North, Ni East, Ei
EC      27+19.681 Bk 103°10'10.0  830.375 5197.419
       27+00 102°02'30.4" 842.904 5182.244
       26+00 96°18'43.9" 896.816 5098.218
       25+00 90°34'57.4" 932.959 5005.157
       24+00 84°51'11.0" 949.894 4906.770
       23+00 79°07'24.5" 946.944 4806.981
BC      22+39.716 75°40'10" 935.576 4747.815


Math check: the coordinates computed for the EC in the table should be the same as the EC coordinates computed from the PI. Within rounding error, that's the case here.

d. Summary

The radial chord method lends itself nicely to computing curve point coordinates. The computations are not complex, although they are admittedly tedious.

Once coordinates are computed, field stakeout is much more flexible using Coordinate Geometry (COGO).