The following are the methods of setting out simple circular curves by linear methods and by the use of chain and tape: 1. By ordinates from the Long chord 2. By Successive Bisection of Arcs. 3. By Offsets from the Tangents. 4. By Offsets from Chords Produced.
Method # 1. By Ordinates from the Long Chord (Fig. 11.8):
Let T1T2=L= the length of the Long chord
ED= O0= the offset at mid-point (e) of the long chord (the versed sine)
PQ=Ox= the offset at distance x from E
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Draw QQ1 parallel to T1 T2 meeting DE at Q1
When the radius of the curve is large as compared with the length of the long chord, the offset may be equated by the approximate formula which is derived as follows:
Note:
In the exact equation (11.1), the distance x of the point P is measured from the mid-point of the long chords; while in the approximate equation (11.11), it is measured from the first tangent point (T1).
Procedure of Setting Out the Curve:
(i) Divide the long chord into an even number of equal parts.
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(ii) Calculate the offsets by the equation 11.10 at each of the points of division.
Note:
1. Since the curve is symmetrical on both sides of the middle- ordinate, the offsets for the right-hand half of the curve are the same as those for the left-hand half.
2. If the offsets are found by the approximate equation (11.11), the long chord should be divided into a convenient number of equal parts and the calculated offsets laid out at each of the points of division.
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This method is used for setting out short curves e.g., curves for street bends.
Method # 2. By Successive Bisections of Arcs (Fig 11.10):
It is also known as Versine Method. Join T1 T2 and bisect it at E. Set out the offset ED the versed since equal to:
Join T1D and DT2 and bisect them at F and G respectively .Then set outsets FH and GK at F and G each equal to 
thus fixing two more points H and K on the curve. Then each of the offsets to be set out at mid points of the chords T1H, HD, DK and KT2 is equal to 
.By repeating this process, set out as many point as are required.
This method is suitable where the ground outside the curve is not favourable to the tangents.
Method # 3. By Offsets from the Tangents:
The offsets may be either radial or perpendicular to the tangents.
(a) By Radial Offsets (Fig 11.11a):
When the radius is large, the offsets may be calculated by the approximate formula, which may be derived as under:
(b) By Offsets perpendicular to the Tangents (Fig 11.11,b):
Procedure of setting out the curve:
(i) Locate the tangent points T1 and T2.
(ii) Measure equal distances, say 15 or 30 m along the tangent from T1.
(iii) Set out the offsets calculated by any of the above methods at each distance, thus obtaining the required points on the curve.
(iv) Continue the process until the apex of the curve is reached.
(v) Set out the other half of the curve from the second tangent.
This method is suitable for setting out sharp curves where the ground outside the curve is favourable for chaining.
Method # 4. By Offsets from Chords Produced (Fig. 11.12):
Let AB = the first tangent; T1 = the first tangent point E, F, G etc on the successive points on the curve T1E = T1E1 = C1 = the first chords.
EF, FG, etc. = the successive chords of length C2, C3 etc., each being equal to the full chord.
∠ BT1E = α in radians = the angle between the tangents BT1 and the first chord T1E.
E1E = O1 = the offset from the tangent BT1
E2F = O2 = the offset from the chord T1E produced.
Produce T1E to E2 such tharEE2 = C2. Draw the tangent DEF1 at E meeting the first tangent at D and E2F at F1.
∠BT1E= α in the radians= the angle between the tangents BT1and the first chord T1E.
E1E=O1= the offset from the tangent BT1
E2F=O2= the offset from the chord T1E produced.
Produce T1E to E2 such that EE2= C2.Draw the tangent DEF1at E meeting the first tangent at D and E2Fat F1.
The formula for the offsets may be derived a under:
∠ BT1E=x
∠T1OE=2x
The angle subtended by any chord at the center is twice the angle between the chord and the tangent
Each of the remaining offsets O4,O5 etc expect the last one (On) is equal to O3. Since the length of the last chord is usually less than the length of the chord, the last offset,
Procedure of Setting out the Curve (Fig. 11.12):
(i) Locate the tangent points (T1 and T2) and find out their changes. From these changes, calculate lengths of first and last sub-chords and find out the offsets by using the equations 11.16 to 11.19.
(ii) Mark a point E1 along the first tangent T1B such that T1E1 equals the length of the first sub-chord.
(iii) With the zero end of the chain (or tape) at T1 and radius = T1E1, swing an arc E1E and cut off E1E = O1, thus fixing the first point E on the curve.
(iv) Pull the chain forward in the direction of T1E produced until the length EE2 becomes equal to the second chord C2.
(v) Hold the zero end of the chain at E. and radius = C2, swing an arc E2F and cut off E2F = O2, thus fixing the second point F on the curve.
(vi) Continue the process until the end of the curve is reached. The last point fixed in this way should coincide with the previously located point T2. If not, find the closing error. If it is large i.e., more than 2 m, the whole curve are moved sideways by an amount proportional to the square of their distances from the tangent point T1. The closing error is thus distributed among all the points.
This method is very commonly used for setting out road curves.