The Tangential Method of Tacheometry
The tangential method in tacheometry is used to calculate horizontal and vertical distances from the instrument to the staff location. These distances are determined based on vertical angles observed from the instrument to two fixed targets on the staff, which are separated by a known distance, denoted as S.
The method can be categorized into three possible scenarios, depending on the nature of the observed vertical angles:
Both vertical angles are elevation angles
In this case, both targets on the staff appear higher than the horizontal line of sight from the instrument.Both vertical angles are depression angles
Here, both targets on the staff are observed below the horizontal line of sight.One angle is an elevation angle, and the other is a depression angle
In this scenario, one target is positioned above the horizontal line of sight while the other is below it.
1. Both vertical angles may be elevation angles.
Consider two targets, A and B, positioned S meters apart on a staff held vertically at point D (refer to Fig. 13.20). Let O represent the trunnion axis of the theodolite. The angles of elevation to the targets A and B are denoted as α₁ and α₂, respectively.
Horizontal Distance Formula:
From triangle ACE, we have:
CE = AE · cot α₁
From triangle BCE, we have:
CE = BE · cot α₂
Equating these two expressions:
AE · cot α₁ = BE · cot α₂
Since AB = S, this becomes:
AE · cot α₁ = (AE + S) · cot α₂
Rearranging terms:
EA · (cot α₁ – cot α₂) = S · cot α₂
Thus, the horizontal distance EA is:
EA = (S · cot α₂) / (cot α₁ – cot α₂) = (S · tan α₁) / (tan α₂ – tan α₁)
Elevation Formula:
The height difference H can be expressed as:
H = AE · cot α₁ = S / (tan α₂ – tan α₁)
Alternatively, H can also be written as:
H = S · (cos α₁ · cos α₂). cosec(α₂ – α₁)
This alternative form is particularly useful for logarithmic calculations.
Relative Level (R.L.) Formula:
The R.L. of D is given by:
R.L. of D = R.L. of the trunnion axis + EA – AD
Substituting for EA:
R.L. of D = R.L. of the trunnion axis + (S · tan α₁) / (tan α₂ – tan α₁) – height of the lower target above D
2. Both vertical angles are depression angles
In this scenario, both vertical angles are depressions. We will use the same notations as in Case I to calculate the horizontal distance and elevation.
Horizontal Distance Formula:
From the geometry of triangles ACE and BCE, we obtain the following relationships:
The height difference can be expressed as:
H = AE cot α₂ = BE cot α₁
By rearranging terms:
(AB + BE) cot α₂ = BE cot α₁
Expanding this equation:
AB cot α₂ + BE cot α₂ = BE cot α₁
Simplifying to solve for BE:
AB cot α₂ = BE (cot α₁ − cot α₂)
Thus, the horizontal distance BE becomes:
BE = S cot α₂ / (cot α₁ − cot α₂)
Or equivalently:
BE = S tan α₂ / (tan α₂ − tan α₁)
Since H = BE cot α₁, the height difference can be expressed as:
H = S / (tan α₂ − tan α₁)
Alternatively, we can also write the height difference as:
H = S cos α₁. cos α₂ . cosec(α₂ − α₁)
Elevation Formula:
The relative level (R.L.) of point D can be determined using the following formula:
R.L. of D = R.L. of the trunnion axis − EB − BD
Relative Level (R.L.) Formula:
Substituting the value for EB:
R.L. of D = R.L. of the trunnion axis − S tan α₂ / (tan α₂ − tan α₁) − height of the upper target above D
3. One of the angles may be an elevation angle and the other may be a depression angle.
In this case, one of the vertical angles is an elevation, and the other is a depression. We will continue using the same notations as in Case I.
Horizontal Distance Formula:
From triangles ACE and BCE, we can derive the following:
H = AE cot α₁ = BE cot α₂
Rearranging the terms:
AE cot α₁ = (S − AE) cot α₂
Expanding and simplifying:
AE cot α₁ = S cot α₂ − AE cot α₂
Further simplification gives:
AE (cot α₁ + cot α₂) = S cot α₂
Solving for AE:
AE = S cot α₂ / (cot α₁ + cot α₂)
Alternatively, we can express this as:
AE = S tan α₁ / (tan α₁ + tan α₂)
Since H = AE cot α₁, the height difference becomes:
H = S / (tan α₁ + tan α₂)
Or, for logarithmic calculations, we can write:
H = S cos α₁ cos α₂ . cosec(α₁ + α₂)
Elevation Formula:
The relative level (R.L.) of point D is determined by:
R.L. of D = R.L. of the trunnion axis − AE − AD
Relative Level (R.L.) Formula:
Substituting for AE:
R.L. of D = R.L. of the trunnion axis − S.tan α₁ / (tan α₁ + tan α₂) − height of the lower target
Disadvantages of the Tangential Method
The tangential method in tacheometry has several drawbacks, including:
- Slower process: It is comparatively time-consuming.
- Complex calculations: The method requires more computations to determine distances and elevations.
- Multiple vertical angle observations: Two vertical angles need to be measured to calculate the distance.
- Risk of instrument disturbance: There is a possibility that the instrument may shift or get disturbed between the observations of the two vertical angles, which may go unnoticed.
Due to these disadvantages, the tangential method is often considered less efficient than the stadia method and is rarely used.
The most commonly used method in tacheometry is the fixed-hair stadia method, which involves using a staff held vertically.
