Circle of Position & Distance Off
Using Distance Circles, Depth Contours, and Vertical Angles for Position Fixing
What Is a Circle of Position?
A circle of position is a circular line of position (LOP) centered on a known point, with a radius equal to the measured distance from that point. If you know you are exactly 3 nautical miles from a lighthouse, your position lies somewhere on a circle of radius 3 NM drawn around that lighthouse on the chart. Just like a bearing LOP, a single circle of position does not fix your position — it only constrains you to be somewhere on the circle. You need a second LOP (another circle, a bearing, or a depth contour) to pin down the exact point.
Circles of position arise from any measurement that gives a distance rather than a direction. The most common sources are radar range readings, vertical sextant angle calculations, and in some cases the rising or dipping range of a light (the distance at which a light appears above or disappears below the horizon). Each of these methods produces a distance that translates directly into a circle on the chart.
In practice, only a small arc of the full circle is relevant — the portion near your estimated position. You do not need to draw the entire circle; a short arc near your DR position is sufficient. However, understanding the full geometry is important for recognizing ambiguous fixes. Two circles can intersect at two points, and you must use your DR position or other information to determine which intersection is your actual location.
You only need to draw the arc of the circle near your estimated position, not the whole circle. Use a drawing compass set to the distance at chart scale to swing the arc quickly.
A circle of position is produced by what type of observation?
Using Depth Contours as LOPs
A depth contour (also called a fathom line or isobath) can serve as a line of position in areas where the bottom topography is distinctive. If your echo sounder reads 10 meters and the chart shows a well-defined 10-meter contour running along the coast, you know your vessel is somewhere on that contour line. This is not a circle in the geometric sense, but it functions as an LOP just like a bearing or distance arc — it constrains your position to lie along a specific line on the chart.
Depth contours are most useful when they run roughly perpendicular to your track and when the bottom slope is steep enough that a small change in position produces a noticeable change in depth. On a gently sloping bottom, the 10-meter contour and the 11-meter contour may be hundreds of meters apart, making it difficult to place your position precisely. On a steep shelf, the contours are tightly packed, and even a one-meter depth change corresponds to a meaningful position shift.
To use a depth contour as an LOP, correct the echo sounder reading for the state of the tide. The charted depth is referenced to chart datum (usually Lowest Astronomical Tide or Mean Lower Low Water), so you must subtract the current tidal height from your sounder reading to compare apples to apples. Alternatively, add the tidal height to the charted depth and compare that to your sounder reading. Either way, the correction is essential — at high tide, the sounder may read several meters deeper than the charted depth at your position.
Always correct echo sounder readings for tidal height before matching them to charted depths. Failure to do so can place you on the wrong contour, especially in areas with large tidal ranges.
Vertical Sextant Angle: Measuring Distance Off
The vertical sextant angle method is the traditional way to measure distance from an object of known height, such as a lighthouse, cliff, or headland. Using a marine sextant, measure the vertical angle between the waterline at the base of the object and its summit (or lantern, in the case of a lighthouse). This angle, combined with the known height of the object, gives the distance by simple trigonometry.
The formula most commonly used is: Distance (NM) = (Height in feet x 0.565) / Angle in minutes of arc. For metric units: Distance (NM) = (Height in meters x 1.854) / Angle in minutes of arc. These are approximations valid for small angles (under about 5 degrees), which covers most practical cases. For greater precision, use Table 9 in Bowditch, which tabulates distance for various heights and angles. Always ensure the height you use is the height above the waterline at the time of observation, not just the charted height — apply a tidal correction if necessary.
Taking a vertical sextant angle requires practice. Set the sextant to zero and sight the base of the object at the waterline. Then slowly increase the angle by turning the micrometer drum until the top of the object (reflected in the index mirror) touches the waterline in the horizon mirror. Rock the sextant slightly to ensure you are measuring the true vertical angle and not an oblique angle. Record the reading and the time, then compute the distance. The resulting distance gives you a circle of position that you can combine with a bearing for a fix.
Practice taking vertical sextant angles to objects of known distance (such as a harbor entrance of known width) to verify your technique and build confidence in the method before relying on it at sea.
Radar Distance Rings
Modern marine radar provides an efficient way to measure distance to identifiable targets. The radar display shows concentric range rings at fixed intervals (for example, every 0.5 NM on the 3 NM range scale). The variable range marker (VRM) is an adjustable ring that can be placed precisely on a target to read its range. Additionally, the cursor on most radar units displays range and bearing when placed over a target echo.
Radar range to a solid, well-defined target — such as a rocky headland, a steep island, or a large buoy with a radar reflector — is typically accurate to 1-3% of range or about 30-50 meters, whichever is greater. This makes radar range one of the most accurate distance measurements available to the coastal navigator. It works equally well in daylight, darkness, fog, and rain (though heavy rain can create clutter that obscures weak targets).
To use radar range for a circle of position, identify the target on both the radar display and the chart. Measure the range using the VRM or cursor. On the chart, set your drawing compass to the measured range at chart scale and swing a short arc centered on the target near your estimated position. Combine this arc with a bearing (from radar, compass, or visual) or with a second radar range to another target for a fix. Two radar ranges to two different targets give two circles of position that intersect at your fix — this is the classic radar fix, which is covered in detail in the next lesson.
Which distance measurement method is typically the most accurate for coastal navigation?
Combining Circles with Bearing LOPs
The most common and practical use of circles of position is in combination with bearing LOPs. A compass bearing to an object gives a straight-line LOP; a distance measurement to the same or a different object gives a circular LOP. The intersection of a line and a circle is geometrically strong — the line cuts the circle at a well-defined point (or two points, one of which is usually obviously wrong), giving a confident fix.
A classic example: you take a compass bearing to a lighthouse and simultaneously measure the radar range to the same lighthouse. The bearing gives a line through the lighthouse; the range gives a circle around it. They intersect at your position. Because the two LOPs are different types (angular vs. distance), their errors are independent — a compass error does not affect the range, and a range error does not affect the bearing. This independence makes the combined fix more reliable than a fix from two bearings, which share the same types of error.
You can also combine a bearing with a depth contour. If a compass bearing to a lighthouse gives an LOP, and your corrected echo sounder reading places you on the 15-meter contour, the intersection of the bearing line with the contour line is your fix. This is particularly useful in approaches to harbors where the bottom topography is well charted and the depth contours are closely spaced. Always check that the contour and the bearing cross at a reasonable angle — if they are nearly parallel, the fix will be imprecise.
When combining a bearing and a distance to the same object, take both measurements as simultaneously as possible. On a vessel moving at 6 knots, a 30-second delay means 50 meters of travel between measurements.
Summary
A circle of position is a circular LOP centered on a known point with a radius equal to the measured distance — the vessel lies somewhere on this circle.
Depth contours serve as LOPs when the echo sounder reading (corrected for tide) matches a well-defined contour on the chart.
The vertical sextant angle method uses the measured angle to an object of known height to calculate distance off.
Radar range provides highly accurate distance measurements, typically within 1-3% of range or 30-50 meters.
Combining a circle of position (distance) with a bearing LOP produces a strong fix because the two error types are independent.
Key Terms
- Circle of Position
- A circular line of position centered on a known point with a radius equal to the measured distance from that point. The vessel's position lies somewhere on this circle.
- Depth Contour (Isobath)
- A line on a chart connecting points of equal depth below chart datum. When corrected for tide, an echo sounder reading that matches a contour constrains the vessel's position to that line.
- Vertical Sextant Angle
- The angle measured with a sextant between the top and base of an object of known height, used to calculate the distance from the object.
- Variable Range Marker (VRM)
- An adjustable range ring on a radar display that can be placed on a target echo to read its distance precisely.
- Chart Datum
- The reference water level to which charted depths are reduced, typically Lowest Astronomical Tide (LAT) or Mean Lower Low Water (MLLW). Actual water depth varies with the tide.