## Objective:

The main objective of this survey job is to set out a horizontal simple circular curve on the ground utilizing two theodolite methods in the field.

## Apparatus:

- Two theodolites
- Two ranging rods

#### Least count of the instrument:

Least count of theodolite = 00°00’05’’

**Related theory:**

#### Principle of two theodolite method

This method is based on the principle that the deflection angle between a tangent and a chord is equal to the angle which the chord subtends in the alternate segment.

#### Mathematical explanation

Suppose, two tangent points are marked on the ground such that T_{1} is the point of commencement and T_{2} be the point of tangency. Let, I be their point of intersection and O is the center of the simple circular curve as shown in the figure:

Let, T_{1 }a be the first sub-chord of length C_{1} and δ_{1} be its deflection angle. As we know from the properties of the circle:

∠ I T_{1 }a = ½ ∠ T_{1}O a

But;

∠ T_{1}T_{2 }a = ½ ∠ T_{1}O a

∠ I T_{1 }a = ∠ T_{1}T_{2 }a = δ_{1}

Similarly, it can be shown that:

∠ I T_{1 }b = ∠ T_{1}T_{2 }b = δ_{2}

#### Practical application of two theodolite method

- The two theodolite method is widely used in setting out railway curves because it gives the correct location of the points.
- This method does not require any tape or chain to fix points on the curve.
- Each point is set irrespective of other points so this method provides more accuracy in fieldwork.
- If any error occurs at one point of the curve, it doesn’t affect the accuracy of other points.
- When it is unfavorable to perform chaining because the ground surface is uneven from undulations, two theodolite method can be very conveniently employed.

## Procedure:

- The following steps are performed to set out a simple circular curve using two theodolite method:
- Mark T
_{1}and T_{2}tangent points on the ground. - Make a table for deflection angles for the first sub-chord, normal chords, and the last sub-chord.
- Center and level the theodolite over T
_{2 }properly. Set vernier A to 0° and tighten the upper clamp. - Release the lower clamp and slowly turn the telescope towards the ranging rod such that the ranging rod at is perfectly T
_{1}bisected. Now fix the lower clamp at this position. - Take off the ranging rod from point T
_{1}. Center and level the second theodolite over this point. Set the vernier A to 0°. First, tighten the upper clamp then release the lower clamp and bisect the ranging rod at B. Again, the lower clamp is fixed is now fixed at this point. - Release the upper clamps of both theodolites and set the first deflection angle δ
_{1}on vernier A of both theodolites. - Locate a point P
_{1}such that the lines of sight of both theodolites intersect at this point. Thus, P_{1}is a point on the curve. - Now, set the next deflection angle δ
_{2}on vernier A of both the instruments. Again, locate a point P_{2}, such that the lines of sight of both instruments intersect at it, hence, P_{2 }is the next point on the curve. - Continue this procedure till all the deflection angles are set out and all the points are marked on the ground.
- Set out the total deflection angle δ
_{n}on both the theodolites such that the line of sight at T_{1}should bisect T_{2}and the line of sight of theodolite at T_{2}should bisect the point B.

## Observations & Calculations:

Points | Chainage (m) | Chord length (m) | Deflection angle δ |

T_{1} |
|||

P_{1} |
|||

P_{2} |
|||

T_{2} |

## Results & Discussions:

A simple circular curve has been set out using two theodolite method.

## Precautions:

- Theodolite should be adjusted and leveled properly and tightly clamped for precise readings.
- Cover theodolite with an umbrella to protect it from direct sunlight and rain.
- Helmet, closed-toe shoes, and safety vests must be worn by all field workers.
- Pegs should be laid vertically by mallet into the ground.

## Precautions:

- In centering and levelling, make sure that the tribrach is made as horizontally as possible.
- Make sure the legs of the stand stay firmly fixed into the ground.
- Theodolite should be adjusted and leveled properly and tightly clamped for precise readings.
- Cover theodolite with an umbrella to protect it from direct sunlight and rain.
- Helmet, closed-toe shoes, and safety vests must be worn by all field workers.
- Follow the proper step-by-step procedure to avoid any errors.
- Establish theodolite station points on firm ground.
- Keep ranging rods exactly vertical by holding it with both hands.
- Place the ranging rods on clean and leveled ground.