A traverse gives you lengths and bearings. But to plot the survey, or to compute areas, or to check that the traverse closes, you need coordinates — and there are two kinds, measured from two different origins.
Getting the distinction clear is essential, because almost every traverse computation depends on converting one into the other.
Latitude and Departure
Before either coordinate type makes sense, we need the two components of a single line.
| Component | Meaning | Axis |
|---|---|---|
| Latitude (L) | The north–south component of a line | y |
| Departure (D) | The east–west component of a line | x |
Any survey line, running in some direction for some distance, can be broken into how far it goes north and how far it goes east. Those two components are the latitude and departure, and together they describe the line completely.
The Sign Convention
| Direction | Sign |
|---|---|
| Northing | Latitude positive |
| Southing | Latitude negative |
| Easting | Departure positive |
| Westing | Departure negative |
This matches the quadrantal bearing system exactly — the first letter (N or S) fixes the sign of the latitude, and the second letter (E or W) fixes the sign of the departure. A line bearing S 30° W has negative latitude and negative departure.
1. Consecutive Coordinates
The coordinates of the end point of a line with respect to its initial point are called consecutive coordinates, or dependent coordinates, of the end point of the line.
So the consecutive coordinates of end point 2 of line 1–2 are simply its latitude and departure — how far north and how far east point 2 lies relative to point 1.
Consecutive coordinates of end point 2 of line 1–2 = −L1 and D1
The negative latitude here simply means that particular line runs southwards.
Why They Are Called “Dependent”
The alternative name is informative. These coordinates depend entirely on where the line started — they say nothing about absolute position, only about the change from one end to the other.
Every line in a traverse has its own consecutive coordinates, and each is measured from a different origin, namely that line’s own starting station. This makes them useless for plotting on their own, but perfect for accumulating.
2. Independent Coordinates
Independent coordinates of a point with respect to a common origin are called independent coordinates.
- Independent coordinates of point 1 = y1 and x1
- Independent coordinates of point 2 = y2 and x2
The Key Phrase: Common Origin
This is what makes independent coordinates useful. Every point is referred to the same origin, so all the coordinates are directly comparable.
You can plot them on a sheet, compute distances between any two stations, calculate areas, and connect the survey to other surveys — none of which is possible with consecutive coordinates alone.
How the Two Are Related
Independent coordinates are built up by accumulating consecutive coordinates:
x2 = x1 + D1
y2 = y1 − L1
x3 = x1 + D1 + D2
y3 = y1 − L1 + L2
Read down the list and the pattern is clear: each new point’s coordinates are the previous point’s coordinates plus the latitude and departure of the connecting line.
The General Rule
Total latitude of any point = Total latitude of the starting point + algebraic sum of the latitudes of all lines up to that point
Total departure of any point = Total departure of the starting point + algebraic sum of the departures of all lines up to that point
The Word “Algebraic” Is Doing Real Work
Note that the sum is algebraic, meaning signs must be respected.
A line running north contributes a positive latitude; a line running south contributes a negative one. If you add magnitudes without signs, a traverse that goes 100 m north and then 100 m south would appear to have travelled 200 m north instead of returning to its starting latitude.
This is exactly why a closed traverse must have latitudes and departures summing to zero — you finish where you began, so all the northings must cancel all the southings, and all the eastings must cancel all the westings.
The Two Coordinate Types Compared
| Consecutive Coordinates | Independent Coordinates | |
|---|---|---|
| Measured from | The line’s own starting point | A common origin |
| Also called | Dependent coordinates | Total coordinates |
| What they give | The change along one line | The absolute position of a point |
| Origin | Different for every line | Same for every point |
| Obtained from | Length and bearing of the line | Accumulating consecutive coordinates |
| Used for | The closure check | Plotting, areas, connecting surveys |
An Analogy That Helps
Think of giving directions. Consecutive coordinates are like saying “walk 200 m north, then 150 m east, then 100 m south” — each instruction is relative to where you currently stand.
Independent coordinates are like giving a grid reference for each corner — every one measured from the same fixed origin, so anyone can find them without following the route.
Both describe the same journey, but only the second lets you plot the result.
Choosing the Origin
When independent coordinates are computed in practice, the origin is chosen deliberately:
The origin is selected such that the entire traverse lies in the north-east quadrant. This is done for ease of plotting the traverse on a sheet, with the left-hand bottom corner of the sheet as the origin.
Why the North-East Quadrant
The reason is practical rather than theoretical. If the origin sits at the bottom-left corner of the drawing sheet, then every point to be plotted lies above and to the right of it — meaning all coordinates come out positive.
Positive numbers are easier to plot, easier to check and far less prone to sign errors than a mixture of positives and negatives. Choosing the origin this way removes an entire category of mistake before it can happen.
Quick Revision Notes
- Latitude is the north–south component of a line (the y direction); departure is the east–west component (the x direction).
- North and east are positive; south and west are negative.
- Consecutive coordinates are the coordinates of the end point of a line with respect to its own initial point. Also called dependent coordinates.
- Independent coordinates are the coordinates of a point with respect to a common origin.
- x2 = x1 + D1 and y2 = y1 − L1.
- Total latitude of any point = total latitude of the starting point + algebraic sum of latitudes of all lines up to that point.
- Total departure of any point = total departure of the starting point + algebraic sum of departures of all lines up to that point.
- The sum is algebraic — signs must be respected.
- The origin is chosen so the whole traverse lies in the north-east quadrant, with the bottom-left corner of the sheet as origin, for ease of plotting.
Mistakes Students Commonly Make
- Swapping latitude and departure. Latitude is north–south; departure is east–west.
- Associating latitude with x. Latitude is the y component; departure is the x component.
- Adding magnitudes instead of taking the algebraic sum, so that southings fail to cancel northings.
- Confusing the two coordinate types. Consecutive coordinates are relative to the line’s own start; independent coordinates are relative to a common origin.
- Forgetting that consecutive coordinates are also called dependent coordinates.
- Choosing an arbitrary origin. It should place the whole traverse in the north-east quadrant so all coordinates are positive.
- Trying to plot directly from consecutive coordinates, which cannot be done because each has a different origin.
Conclusion
Every traverse line breaks into a latitude and a departure — how far north and how far east it runs. Taken relative to the line’s own starting point, those are the consecutive or dependent coordinates, and they describe change rather than position. Accumulate them algebraically from a single fixed origin and they become independent coordinates, which give the absolute position of every station and can therefore be plotted, measured and connected to other surveys. Choosing that origin so the whole traverse falls in the north-east quadrant keeps every number positive and every plot straightforward.
Frequently Asked Questions
What is latitude and departure of a line?
Latitude is the north–south component of the line, corresponding to the y direction, and departure is the east–west component, corresponding to the x direction.
What are consecutive coordinates?
The coordinates of the end point of a line with respect to its initial point. They are also called dependent coordinates.
Why are consecutive coordinates called dependent?
Because they depend entirely on where that particular line started, describing only the change along the line rather than any absolute position.
What are independent coordinates?
The coordinates of a point with respect to a common origin shared by every point in the survey.
How are independent coordinates calculated?
By accumulating consecutive coordinates. For example x₂ = x₁ + D₁ and y₂ = y₁ − L₁, with each further point adding the latitude and departure of the connecting line.
What is the rule for total latitude?
Total latitude of any point equals the total latitude of the starting point plus the algebraic sum of the latitudes of all lines up to that point.
Why must the sum be algebraic?
Because northings are positive and southings negative. Adding magnitudes without signs would make a traverse that returns to its starting latitude appear to have travelled a long way north.
Why is the origin chosen so the traverse lies in the north-east quadrant?
So that all coordinates come out positive, which makes plotting on a sheet easier and removes a common source of sign errors. The bottom-left corner of the sheet is used as the origin.
Which coordinates are used for plotting?
Independent coordinates, because they all share a common origin. Consecutive coordinates cannot be plotted directly since each is measured from a different starting point.