CONCEPTS OF SINGLE PLANE METHOD

 


 

The "Single Plane Method" is primarily used in surveying for trigonometric levelling to determine the elevation of an inaccessible object, like a building's top. It involves setting up a theodolite at two points on the ground, measuring the angle of elevation to the object from both points, and taking staff readings at each station. By using the observed angles, the known horizontal distance between the stations, and the staff readings, trigonometric calculations are used to find the object's horizontal distance and height. 

Key Concepts & Procedure 

1.     Triangulation/Same Plane Assumption: 

The core idea is that the two instrument stations and the object are in the same vertical plane.

2.     Measurements:

·        Station 1 (A): Set up the theodolite, ensure it's level, and record the staff reading (S1) for the line of sight parallel to the ground (zero line of sight). Then, measure the angle of elevation (α1) to the top of the object.

·        Station 2 (B): Move the theodolite to a second point (B) a known distance (d) away from station A. Take a staff reading (S2) and measure the new angle of elevation (β1) to the object's top.

3.     Calculations:

·        Height above line of sight: Use trigonometry (tan α1 = H1/D and tan β1 = H2/(D+d)) to find the height of the object above the line of sight at each station (H1 and H2).

·        Total Height: Add the staff reading to the height above the line of sight (Total Height = S1 + H1) to get the object's final height.

·        Distance: Solve for the unknown horizontal distance (D) from the first station to the object's base.

Purpose 

·        To find the elevation of high and inaccessible objects such as tall buildings, towers, or chimneys.

·        To calculate the horizontal distance to the object when the distance is not easily measurable.

 Cases for the single plane method in trigonometric leveling depend on the accessibility of the object's base and the position of the instrument station relative to the elevated object. The three primary cases are: (1) Base accessible: both horizontal distance and height are measured directly. (2) Base inaccessible, instrument stations in the same vertical plane as the object: the instrument station is either at the same level or a different level from the object's base. (3) Base inaccessible, instrument stations not in the same vertical plane as the object: the most complex case, requiring two instrument stations to determine the object's elevation.  

Case 1: Base of the object is accessible

·        Procedure: 

You set up the theodolite at one point and measure the angle of elevation to the top of the object, along with the staff reading at a base point. 

·        Calculation: 

You then use trigonometric formulas to determine the horizontal distance from the instrument to the object and its elevation. 

Case 2: Base of the object is inaccessible, and the instrument station is in the same vertical plane as the object 

·        Procedure: 

This method is used when you cannot directly measure the horizontal distance to the object. You set up the theodolite at two different stations (e.g., A and B), which are in the same vertical plane as the elevated object.

·        Sub-Cases:

·        Instrument axis at the same level: The height of the object is determined using the angles of elevation and the known distance between the instrument stations.

·        Instrument axis at a different level: This involves a more complex formula that accounts for the difference in height between the instrument stations.

Case 3: Base of the object is inaccessible, and the instrument stations are not in the same vertical plane as the elevated object 

·        Procedure: 

This is the most intricate case, where the instrument stations and the object do not share the same vertical plane.

·        Calculation: 

You use more complex trigonometric formulas, or sometimes a two-station approach, to solve for the object's elevation.

 

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