The following two methods are the methods of setting out simple circular curves by angular or instrumental methods: 1. By Rankine’s Tangential Angles. 2. By Two Theodolites.
Method # 1. Rankine’s Method of Tangential or Deflection Angles: (Fig. 11.14):
In this method, the curve is set out by the tangential angles (also known as deflection angles) with a theodolite and a chain (or tape). The method is also called as chain and theodolite method.
The deflection angles are calculated as follows:
Let T1 and T2 be the tangent points and AB the first tangent to the curve.
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D, E, F, etc. =the successive points on the curve,
R = the radius of the curve.
C1, C2, C3 etc. = the lengths of the chords T1D, DE, EF etc., i.e., 1st, 2nd, 3rd chords etc.
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δ 1, δ 2, δ 3 etc. = the tangential angles which each of the chords T1 D1, DE, EF, etc., makes with the respective tangents T1, D, E. etc.
∆1, ∆2, ∆ 3 etc. = the total tangential or deflection angles which the chords T1D, DE, EF, etc. make with the first tangent AB.
Since each of the chord lengths C2, C3, C4…………. Cn-1 is equal to the length of the full chord, δ2 = δ3 = δ4………….. δ n-1.
It is well known preposition of geometry that the angle between the tangent and a chord equals the angle which the chord subtends in the opposite segment.
Now ∠DT1E is the angle subtended by the chord DE in the opposite segment, therefore, it is equal to the tangential angle (δ2) between the tangent D and the chard DE
Check:
The total deflection angle BT1 T2 
where φ is the deflection angle of the curve.
If the degree of die curve (D) is known, the deflection angle for 30 m chord is equal 1/2D degrees, and that for the sub-chord of length C1,
Procedure of Setting out the Curve:
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(i) Locate the tangent points (T1 and T2) and find out their changes. From these changes, calculate the lengths of first and last sub-chords and the total deflection angles for all points on the curve as described above.
(ii) Set up and level the theodolite at the first tangent point (T1).
(iii) Set the vernier A of the horizontal circle to zero and direct the telescope to the ranging rod at the intersection point B and bisect it.
(iv) Loosen the vernier plate and set the vernier A to the first deflection angle Δ1, the telescope is thus directed along T1D. Then along this line, measure T1D equal in length to the first sub-chord, thus fixing the first point D on the curve.
(v) Loosen the upper clamp and set the vernier A to the second deflection angle Δ2, the line of sight is now directed along T1E. Hold the zero end of the chain at D and swing the other end until the arrow held at that end is bisected by the line of sight, thus fixing the second point (E) on the curve.
(vi) Continue the process until the end of the curve is reached. The end point thus located must coincide with the previously located point (T2). If not, the distance between them is the closing error. If it is within the permissible limit, only the last few pegs may be adjusted; otherwise the curve should be set out again.
Note:
In the case of a left-handed curve, each of the values Δ1, Δ 2 Δ 3 etc, should be subtracted from 360° to obtain the required value to which the vernier is to be set i.e. the vernier should be set to (360° – Δ1), (360° – Δ2), (360° – Δ 2) etc. to obtain the 1st, 2n, 3rd etc, points on the curve.
This method gives highly accurate results and is most commonly used for railway and other important curves.
Method # 2. Two-Theodolite Method (Fig. 11.16):
This method is very useful in the absence of chain or tape and also when ground is not favourable for accurate chaining. This is simple and accurate method but requires essentially two instruments and two surveyors to operate upon them, so it is not as commonly used as the method of deflection angles. In this method, the property of circle ‘that the angle between the tangent and the chord equals the angle which that chord subtends in the opposite segment’ is used.
Let D, E, F, etc. be the points on the curve. The angle (Δ1) between the tangent T1B and the chord T1D i.e. ∠BT1 D = ∠T1T2D. Similarly, ∠BT1E = ∆2 = ∠T1T2 E, and ∠BT1F = ∆3 = ∠T1T2F etc. The total deflection angles ∆1, ∆2, ∆3, etc. are calculated from the given data as in the first method (i.e. as in Rankine’s method of deflection angles).
Procedure of setting out the curve:
(i) Set up two theodolites, one at T1 and the other at T2.
(ii) Set vernier of the horizontal circle of each of the theodolites to zero.
(iii) Turn the instrument at T1 to sight the intersection point B and that at T2 to sight T1.
(iv) Set the vernier of each of the instruments to read the first deflection angle Δ1. Now the line of sight of the instrument at T1 is along T1D and that of the instrument at T2 is along T2D. Their point of intersection is the required point on the curve Direct the assistant to move the ranging rod until it is sighted exactly by both the theodolites, thus fixing the point D on the curve.
(v) Then set the vernier of each of the instrument to the second deflection angle Δ2, proceed as before to obtained the second point (E) on the curve.
(vi) Repeat the process until the whole curve is set out.
Note:
It may so happen that the point T1 may not be visible from the point T2. In such a case, direct the telescope of the instrument at T2 towards B with the vernier A set to zero. Now loosen the vernier plate and set the vernier A to read an angle of 
. The telescope is thus directed along T2 T1. For the first point D on the curve, set the vernier A to read 
. Similarly for the second point E, set the vernier A to
, and so on.