Section 3. Radial Chord Method
a. Circlular Geometry
For any circular arc, the angle between the tangent at one end of the arc and the chord is half the arc's central angle, Figure C21.
Figure C21 Deflection angle 
Angle a/2 is the deflection angle from one end of the arc to the other. The chord's length is computed from:
Equation C10 
In terms of the degree of curvature, Figure C22:
Figure C22 Full station deflection angle 
The deflection angle for a full station is half the degree of curvature. Since the deflection angle is D/2 and it occurs over a 100.00 ft, the deflection rate can be computed from:
Equation C11 
Extending this geometry to the entire curve, Figure C23, the total deflection angle at the BC from the PI to the EC is Δ/2.
Figure C23 Deflection angle for entire curve 
Since the deflection angle occurs across the curve's length, the deflection rate can also be written as Equation C12.
Equation C12 
b. Radial chords
One way to stake a horizontal curve is by the radial chord method, Figure C24.
Figure C24 Radial chord method 
An instrument is set up on the BC and the PI sighted as a backsight. Then to stake each curve point, a defelction angle is turned and chord distance measured. For each point to be staked, we need to compute its deflection angle and chord distance.
The deflection angle to any point i on the curve, Figure C25, can be computed from Equation C13.
Figure C25 Deflection angle and radial chord 
Equation C13 
l_{i} is the arc distance to the point from the BC and is computed using Equation C14.
Equation C14 
Equation C10 can be rewritten using the deflection angle:
Equation C15 
Using Equations C13 to C15, the defelction angle and distance any curve point from the BC can be computed.
c. Example
Determine the radial chord stakeout data at full stations for the example from Section 2.d.
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 
Use Equation C11 to compute the curve's deflection rate:
Set up Equations C13, C14, and C15 for the curve:
Set up the Curve Table:
Curve Point 
Arc dist, l_{i}, (ft) 
Defl angle,δ_{i} 
Radial chord, c_{i} 

EC  27+19.681 Bk 



27+00 




26+00 




25+00 




24+00 




23+00 




BC  22+39.716 
Solve the three equations for each curve point and record the results in the table.
At 22+39.716, we're still at the BC so all three entries are zero.
At 23+00:
And so on until the table is complete.
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 
Math checks:
Arc distances between successive full stations differ by 100.00 ft.
At the EC:
Arc distance should equal the curve length
Defl angle should equal Δ/2
Radial chord shoud equal Long Chord
The difference between deflection angles at successive full staions should be D/2.
These checks have been met.
Remember that we carried additonal decimal places to minimize rounding errors. Rounding can become quite pronounced for long flat curves so computation care must be exercised. Once the table has been computed, the final values can be shown to reasonable accuracy levels. For example, distances can be shown to 0.01 ft and deflection angles to 01".
d. Summary
Computing and staking curve points by the radial chord method is simple and straightforward. Using the curve equations, any point on the curve can be computed and staked, not just those included in the curve table.
However, in the field, it may not be the most efficient way to stake a curve using modern instrumentation, particularly with long curves which may have chords thousands of feet long at very small deflection angles. Most contemporay survey computations and fieldwork use coordinates giving greater stakeout flexibility. While the radial chord method may not be used for stake out, it is well adapted to coordinate computations as we'll see in the next section.