how to find missing line distance in a traverse?

Understanding the Missing Line Problem

A missing line problem occurs when the length and bearing of a side in a traverse are unknown. This can happen due to physical obstructions, environmental factors, or data corruption. Solving such problems requires the use of mathematical formulas to calculate the unknowns based on the available data for the other sides.

This guide will walk through:

• The theory behind departures and latitudes.
• The steps to solve for the missing line in a closed traverse.
• A worked example to clarify the process.

Key Concepts: Departure and Latitude

In surveying, the departure and latitude of a line represent its east-west and north-south components, respectively. These are calculated using the line’s length ("L") and bearing ("θ"):

• Departure (∆X): ∆X = L × sin(θ)
• Latitude (∆Y): ∆Y = L × cos(θ)

The bearing determines the direction of the line relative to a reference meridian, typically measured clockwise from north. The quadrant of the line’s bearing affects the signs of the departure and latitude:

• Quadrant I: Both ∆X and ∆Y are positive.
• Quadrant II: ∆X is positive, ∆Y is negative.
• Quadrant III: Both ∆X and ∆Y are negative.
• Quadrant IV: ∆X is negative, ∆Y is positive.

By summing the departures and latitudes of all lines in a closed traverse, we can calculate the components of the missing line.

Step-by-Step Solution for One Missing Line

Step 1: Organize the Data

Create a table that includes the following columns for each line in the traverse:

• Line name (e.g., AB, BC, CD, etc.).
• Length (L).
• Bearing (θ).
• Departure (∆X = L × sin(θ)).
• Latitude (∆Y = L × cos(θ)).

Step 2: Calculate Departures and Latitudes for Known Lines

For each line with known length and bearing, compute the departure and latitude using the formulas above. Pay attention to the quadrant to ensure the correct signs.

Step 3: Sum the Departures and Latitudes

In a closed traverse, the algebraic sum of departures and latitudes should equal zero. Therefore, the departure and latitude of the missing line are:

• Departure of missing line: ∆X_m = −(∆X₁ + ∆X₂ + ∆X₃ + …)
• Latitude of missing line: ∆Y_m = −(∆Y₁ + ∆Y₂ + ∆Y₃ + …)

Step 4: Calculate the Missing Line’s Length and Bearing

Once the departure (∆X_m) and latitude (∆Y_m) of the missing line are determined:

• Length ("L_m"): Use the Pythagorean theorem:
• Bearing ("θ_m"): Use the inverse tangent function:

Consider the quadrant of the missing line to adjust the bearing correctly.

Worked Example

Let’s solve a problem where one side of a closed traverse is missing. The traverse is named ABCDEA, and line EA is the missing side. The known data for the traverse are as follows:

Line	Length (m)	Bearing (°)
AB	290	118
BC	320	45
CD	250	210
DE	180	300

Step 1: Compute Departures and Latitudes

For each line, calculate the departure (∆X) and latitude (∆Y):

Line	Length (m)	Bearing (°)	Departure (∆X)	Latitude (∆Y)
AB	290	118	256	−136
BC	320	45	226	226
CD	250	210	−129	−216
DE	180	300	156	−90

Step 2: Sum Departures and Latitudes

Sum the departures and latitudes for all known lines:

• Total departure: …
• Total latitude: …

The missing line EA must balance these sums to zero:

• Departure of EA: …
• Latitude of EA: …

Step 3: Calculate Length and Bearing of EA

• Length: …
• Bearing: …

Conclusion

Solving missing line problems in traverses is a fundamental skill for surveyors. By systematically calculating departures and latitudes, summing them, and applying geometric formulas, the unknowns can be determined efficiently. Mastery of these methods ensures accurate mapping and minimizes errors in surveying projects.

This article covered the solution for one missing line; future guides will delve into scenarios involving multiple missing sides. Stay tuned for more advanced techniques and real-world applications!