Bearing & Distance Fix
Fixing Position with One Bearing and One Distance Measurement
The Concept: Bearing Plus Distance
A bearing and distance fix combines two different types of lines of position. The compass bearing to a charted object gives a straight-line LOP radiating from that object. The distance from the same (or another) object gives a circle of position — a circle centered on the object with a radius equal to the measured distance. The intersection of the bearing line with the distance circle fixes the vessel's position. Since a straight line can intersect a circle in at most two points, and one of those points is usually ashore or otherwise impossible, the fix is typically unambiguous.
This technique is especially valuable when only one suitable landmark is available for bearings. Rather than needing two or three separate objects spread around the horizon, you take a bearing and measure a distance to the same object. The resulting fix can be just as accurate as a two-bearing fix, and in some cases more so, because the bearing line and the distance circle always cross at a strong angle near the object.
Distance can be measured by several methods: radar range is the most direct and accurate, but traditional techniques such as the vertical sextant angle (using the known height of a lighthouse or cliff) and the stadimeter are still valuable skills. Even a rough distance estimate from the known visibility range of a light can provide a useful circle of position in some circumstances.
When using bearing and distance to the same object, take the bearing and distance as simultaneously as possible. On a moving vessel, even a short delay introduces error.
What type of LOP does a distance measurement from a single object produce?
Measuring Distance Off
The vertical sextant angle method is the classic way to measure distance off a landmark of known height. Using a sextant, measure the vertical angle between the waterline at the base of the object and its top (a lighthouse lantern, cliff summit, etc.). Then use the formula: Distance (nautical miles) = height (feet) x 0.565 / angle (minutes of arc), or consult Table 9 in Bowditch. The object's height must be its charted height above the water level at the time of observation, so apply any tidal correction if the charted height is referenced to a different datum.
Radar range is the most convenient modern method. Aim the electronic bearing line (EBL) at the target and read the range from the variable range marker (VRM) or the cursor readout. Radar range to a distinct target — a rocky headland, an island, a large buoy with a radar reflector — is typically accurate to within 1-3% of the range, making it one of the best distance measurements available. Radar range is especially valuable in fog or at night when visual distance methods are impossible.
For objects of known dimensions, you can also estimate distance by the angle the object subtends. If you know a lighthouse is 20 meters tall and it subtends 1 degree in your binoculars (which have a reticle or known field of view), geometry gives the distance. In practice, this is a rougher method than the sextant angle or radar, but it can provide a useful sanity check or a supplementary LOP when other methods are not available.
When using the vertical sextant angle method, remember that charted lighthouse heights are usually given above Mean High Water Springs (MHWS). At low tide, the actual height above the waterline will be greater, and your calculated distance will be slightly too small (placing you closer than you really are). Always apply the tidal correction.
The Bow-and-Beam Bearing Method
The bow-and-beam bearing method is a practical technique for finding both your distance off an object and your position, using two bearings taken at different times. Take the first bearing when the object is approximately 45 degrees on the bow (relative bearing 045 or 315). Record the time and log reading. Continue on course until the object is abeam (relative bearing 090 or 270) and take the second bearing. Record the time and log again.
The geometry of a 45-90 degree triangle means that the distance run between the two bearings equals the distance off the object when it is abeam. If you traveled 2.3 nautical miles between the first and second bearings, then you are 2.3 nautical miles from the object when it comes abeam. This gives you both a bearing (abeam, so roughly 90 degrees to your course) and a distance at the moment of the second bearing — a complete fix.
This method requires that you maintain a steady course and know your speed over the ground accurately. If there is a current setting you sideways, or if your speed is uncertain, the distance calculation will be off. Correct for current if possible by using your course and speed made good rather than your course and speed through the water. Despite its simplicity, the bow-and-beam bearing method is remarkably useful on coastal passages when only one landmark is visible.
You do not need to wait for exactly 45 degrees on the bow. Any pair of angles where the second is the beam (90 degrees) works — just use trigonometry or the appropriate table to find the distance off.
Doubling the Angle on the Bow
Doubling the angle on the bow is the generalized version of the bow-and-beam method. Take a bearing when the object is at some angle on the bow — say 30 degrees. Continue on course and take a second bearing when the angle on the bow has doubled — in this case, 60 degrees. The geometry of an isosceles triangle guarantees that the distance run between the two bearings equals the distance from the object at the time of the second bearing.
This technique works for any initial angle, as long as the second angle is exactly double the first. Common pairs are 22-44, 25-50, 30-60, and the special case 45-90 (the bow-and-beam method). The smaller the initial angle, the longer you wait between bearings, which increases the error from uncertain speed or current. Larger initial angles give shorter intervals but may be harder to judge precisely. An initial angle of 30 degrees (doubling to 60) is a popular compromise.
As with the bow-and-beam method, accuracy depends on maintaining a steady course and knowing your speed made good. Record the log reading at each bearing, and if you suspect a cross-current, apply a correction for set and drift. The fix you obtain is valid at the time of the second bearing only — by the time you have plotted it, you have moved on. Transfer the fix forward to your current position using your DR if necessary.
In the bow-and-beam bearing method, what does the distance run between the two bearings equal?
Summary
A bearing and distance fix uses a compass bearing (straight-line LOP) combined with a distance measurement (circle of position) to fix the vessel's position.
Distance can be measured by radar range, vertical sextant angle to an object of known height, or other methods.
The bow-and-beam bearing method uses bearings at 45 and 90 degrees on the bow; the distance run between them equals the distance off when abeam.
Doubling the angle on the bow is the generalized technique — any initial angle doubled gives the same isosceles-triangle relationship.
All distance-run methods require accurate knowledge of course and speed made good, and should be corrected for current when possible.
Key Terms
- Circle of Position
- A circle on the chart centered on a known point with a radius equal to the measured distance. The vessel lies somewhere on this circle.
- Vertical Sextant Angle
- The angle measured with a sextant between the top of an object and its waterline base, used with the object's known height to calculate distance off.
- Bow-and-Beam Bearing
- A technique using bearings taken when an object is 45 degrees on the bow and again when abeam (90 degrees), where the distance run equals the distance off at the beam bearing.
- Doubling the Angle on the Bow
- A technique where a second bearing is taken when the angle on the bow is double the first angle, creating an isosceles triangle where the distance run equals the distance off at the second bearing.
- Distance Off
- The perpendicular or direct distance from the vessel to a landmark or navigational feature, as opposed to the distance along the vessel's track.