4. Curves and Coordinates
a. Coordinate Equations
Equations C16 and C17 are general equations for computing coordinates using direction and distance from a known point, Figure C26.
Equation C16  
Equation C17 
Figure C26. 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 C27 
The original tangent lines have directions; PC has coordinates.  
Figure C28 
A curve is fit to the tangents. End points are at distance T from the PI along the tangents. 

Figure C29 
Compute coordinates of BC using backdirection of the tangent BCPI and T.


Figure C30 
Compute coordinates of EC using direction of the tangent PIEC and T. These will be use for a later math check.


Figure C31 
Use a curve point's deflection angle to compute the direction if its radial chord from the BC.
δ is positive for right deflections, negative for left.

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 C21 through C24 for this curve.
This is the Radial Chord table computed previously:
Curve Point 
Arc dist, li, (ft) 
Defl angle,δi 
Radial chord, ci 

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, Az_{i}  North, N_{i}  East, E_{i}  
EC  27+19.681 Bk  
27+00  
26+00  
25+00  
24+00  
23+00  
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, Az_{i}  North, N_{i}  East, E_{i}  
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 Coodrinate Geometry (COGO).