A traverse is the workhorse of modern surveying. Rather than building a rigid network of triangles across a landscape, you simply walk from station to station, measuring how far you went and which way you turned.
This post covers what a traverse is, why it displaced triangulation, and a formula that answers a question surveyors must always ask: how accurately do I need to measure distance, given how accurately I am measuring angles?
What a Traverse Is
A traverse is a series of connected lines whose length and direction are measured in the field.
Note the two words length and direction. This is exactly the pairing introduced in compass surveying — one distance and one bearing per line is enough to fix each new station from the last. A traverse is simply that idea extended into a connected chain.
The Five Stages of Field Work
The field work in a theodolite traverse consists of:
- Reconnaissance
- Selection and marking of stations
- Measurement of traverse lines
- Angular measurements
- Picking up the details
Why Reconnaissance Comes First
The order is not arbitrary. Reconnaissance means walking over the area before measuring anything, to see the ground and decide where the stations should go.
This matters because a badly placed station cannot be fixed later. Stations need to be intervisible, on firm ground, safe from traffic, and positioned so that the lines between them can actually be measured. Deciding all of that after you have started measuring means starting again.
It is the principle of working from whole to part applied to fieldwork planning — settle the overall framework first, then fill in the detail.
Details Come Last
Equally, picking up the details is the final stage. The traverse framework is established and checked first; only then are the buildings, fences and other features surveyed relative to it. Again — whole to part.
What a Traverse Is For
A theodolite traverse is commonly used for providing a horizontal control system to determine the relative positions of various points on the surface of the earth.
The phrase horizontal control connects back to the control points discussed in the fundamentals chapter — points of known coordinates from which other measurements are taken. A traverse is how such a framework is established over a working area.
Why Traversing Replaced Triangulation
- Earlier, when sophisticated distance measurement instruments were not available, surveyors relied on triangulation — a method in which one base line was measured and all angles were measured to find the lengths of other lines.
- However, with the advent of Electronic Distance Measurement Instruments (EDMI), traversing is fast replacing triangulation.
The Reason Behind the Shift
This is a genuinely interesting piece of surveying history, and the logic is worth following.
Triangulation existed because measuring long distances accurately was very hard. Dragging a tape across kilometres of rough country, correcting for sag, temperature, pull and slope, was slow and error-prone. Measuring angles, by contrast, was comparatively easy with a good theodolite.
So triangulation minimised the hard part: measure one base line with enormous care, then compute every other length from angles alone.
EDM changed the economics completely. Suddenly a long distance could be measured in seconds, to high precision, by pressing a button. The constraint that triangulation was designed around simply vanished — and once distances became easy, the simpler and more flexible traverse became the better method.
This is the same reversal that produced trilateration, where all three sides of a triangle are measured with EDM and the angles are computed instead.
Classification of Traverses
| Type | Characteristic |
|---|---|
| Closed traverse | Returns to its starting point, or ends on a known control point |
| Open traverse | Does not return or close on a known point |
The distinction matters enormously for accuracy. A closed traverse checks itself — the latitudes and departures must sum to zero, and the angles must sum to a known total. An open traverse has no such check, so an error made anywhere along it will never be detected.
This is why open traverses are used mainly for long, narrow projects like roads and canals, where closing back is impractical, and why they demand extra care.
Methods of Traversing
Two kinds of measurement are involved:
- Linear measurement
- Angular measurement
Linear Measurement Can Be Done By
- Taping or chaining
- Tacheometric method
- Electronic Distance Measurement Instrument (EDMI)
These three represent an ascending order of sophistication — physical contact with the ground, optical measurement from a distance, and finally electronic measurement.
Matching Linear Precision to Angular Precision
This is the most important idea in the topic, and it is frequently examined.
The linear measurement equipment chosen should be such that the degree of accuracy in it is of the same order as that of the angle measurement instrument.
In other words, if very precise equipment is used for angular measurement, we must use equally precise equipment for distance measurement.
Why Mismatched Precision Is Wasted Effort
Think about what happens if the two do not match.
Suppose you measure angles with a one-second optical theodolite but pace out the distances. The final position of each station depends on both quantities — and it will be no better than the worse of the two. All the care taken over the angles is thrown away by the crude distances.
The reverse is equally wasteful. Measuring distances to millimetre precision with EDM while reading angles off a compass wastes the EDM.
A chain is only as strong as its weakest link — and in a traverse, the position of every station is the product of an angle and a distance.
The Precision Relationship
tan(δθ) = δl / l
or for small angles, δθ = δl / l
This relationship is used to determine the consistent precision in angular and linear measurement.
Where It Comes From
Picture a line of length l. An angular error of δθ at one end swings the far end sideways by an arc of approximately l × δθ.
For that sideways displacement to be no worse than the error in the length itself, we need it to equal δl — giving l δθ = δl, or δθ = δl/l.
So the formula simply states that the sideways error from the angle should match the endwise error from the distance. Neither dominates, and neither is wasted.
Worked Example 1: Compass with 30′ Least Count
If a compass with a least count of 30 minutes is used, the linear measurement equipment must have an error of the order of:
δl/l = tan(30′) = tan(0.5°) = 0.008727
= 1 in 115
So a compass traverse needs distances accurate to only about one part in 115 — roughly 9 cm in every 10 m. Ordinary chaining easily achieves this, which is why compass surveying and chaining go together naturally.
Worked Example 2: Theodolite with 20″ Accuracy
If angular measurements are made with an error of 20 seconds, the relative accuracy of the linear measurements is:
δl/l = tan(20″) ≈ 20″ in radians
= (20 / (60 × 60)) × (π / 180) radian
= 1 / 10300
Hence distance should be measured with a precision of 1 in 10,000.
Comparing the Two Examples
| Instrument | Angular Accuracy | Required Linear Precision |
|---|---|---|
| Compass | 30′ | 1 in 115 |
| Vernier theodolite | 20″ | 1 in 10,000 |
The contrast is striking. Improving angular accuracy from 30 minutes to 20 seconds — a factor of 90 — demands that distance accuracy improve by the same factor, from 1 in 115 to about 1 in 10,000.
That is precisely why a theodolite traverse cannot be chained casually. It needs a steel band with all the tape corrections applied, or better still an EDM instrument. The angular precision of the theodolite obliges a matching effort on the distances.
The Radian Conversion
Note the step used to convert seconds into radians, since it appears in many numericals:
Angle in radians = (seconds / 3600) × (π / 180)
For small angles, the tangent of an angle equals the angle in radians, which is why tan(20″) can be replaced by 20″ expressed in radians.
Quick Revision Notes
- A traverse is a series of connected lines whose length and direction are measured in the field.
- Field work stages: reconnaissance, selection and marking of stations, measurement of traverse lines, angular measurements, picking up the details.
- A theodolite traverse provides a horizontal control system.
- Triangulation measured one base line and all angles; EDMI has made traversing replace triangulation.
- Traverses are closed or open.
- Linear measurement by taping or chaining, tacheometric method, or EDMI.
- Linear and angular precision must be of the same order.
- tan(δθ) = δl/l, or δθ = δl/l for small angles.
- Compass with 30′ least count → linear precision 1 in 115.
- Angular error of 20″ → linear precision 1 in 10,000.
Mistakes Students Commonly Make
- Forgetting to convert the angle to radians before equating it to δl/l.
- Confusing minutes and seconds. 30′ gives 1 in 115; 20″ gives 1 in 10,000.
- Assuming higher angular precision alone improves a survey. It is wasted unless distance precision matches.
- Saying triangulation measures all sides. It measures one base line and computes the rest from angles.
- Listing fewer than five stages of field work, usually omitting reconnaissance.
- Omitting the tacheometric method from the linear measurement options.
- Thinking an open traverse can be checked internally. It cannot — only a closed traverse checks itself.
Conclusion
A traverse strings together lines whose length and direction are both measured, and it has largely replaced triangulation because EDM removed the difficulty that made triangulation necessary. The key principle is balance: since each station’s position depends on both an angle and a distance, the two measurements must be of comparable precision or the better one is wasted. The relationship tan δθ = δl/l makes that quantitative — a 30-minute compass needs distances good to 1 in 115, while a 20-second theodolite demands 1 in 10,000.
Frequently Asked Questions
What is a traverse?
A series of connected lines whose length and direction are measured in the field.
What does the field work in a theodolite traverse consist of?
Reconnaissance, selection and marking of stations, measurement of traverse lines, angular measurements, and picking up the details.
What is a theodolite traverse used for?
To provide a horizontal control system for determining the relative positions of various points on the surface of the earth.
Why has traversing replaced triangulation?
Because triangulation existed to avoid measuring long distances, which was once very difficult. Electronic Distance Measurement Instruments made accurate distance measurement fast and easy, removing the constraint that triangulation was designed around.
How can linear measurements be made in a traverse?
By taping or chaining, by the tacheometric method, or by using an Electronic Distance Measurement Instrument.
Why must linear and angular precision match?
Because the position of each station depends on both a distance and an angle, so the result is only as good as the worse of the two. Precision spent on one is wasted if the other is coarse.
What is the relationship between angular and linear precision?
tan(δθ) = δl/l, or for small angles δθ = δl/l, where δθ is the angular error in radians and δl/l is the relative linear precision.
What linear precision is needed with a compass of 30 minutes least count?
About 1 in 115, since tan(30 minutes) equals approximately 0.008727.
What linear precision is needed for an angular error of 20 seconds?
About 1 in 10,300, so distances should be measured to a precision of 1 in 10,000.
How do you convert seconds of arc into radians?
Divide by 3600 to get degrees, then