Position lines and fix geometry

What does one observation actually constrain?

A single line of position usually leaves motion along that line unresolved.

One observation normally leaves a locus

A bearing constrains the aircraft to a ray or line relative to a known station. A horizontal range constrains it to a circle. A visual transit aligns two identifiable features into a line of position. A single such observation usually does not provide a unique two-dimensional position. Identify what the instrument or sighting actually measures before drawing a fix.

Two independent position lines can intersect at a fix. Simultaneous observations avoid the additional motion assumption needed when one line is older. Station identity, feature identity, measurement reference and units must all be correct. An accurate bearing plotted from the wrong station is still a wrong position line.

Crossing angle controls sensitivity

Near-parallel lines intersect at a poorly conditioned angle: a small shift in one line can move the apparent fix a long way. With comparable independent errors, an angle near 90° is generally much better conditioned. A visually neat intersection does not prove accuracy.

In a simplified one-error model, a perpendicular line displacement e creates an intersection shift e/|sin θ|. At 30°, the amplification factor is 2; at 10°, it is 5.76. This is a geometry illustration, not a guaranteed containment bound for a complete navigation system. Shared and systematic errors require separate treatment.

Compare two fixes with the same measurement error

Let one line have a 1 NM perpendicular error. At 90° crossing, the intersection moves 1 NM; at 10°, it moves about 5.76 NM. Reducing the measurement error to 0.5 NM halves each result, but does not repair the poor geometry. Use the laboratory to distinguish improving a sensor measurement from choosing better station or feature geometry. Two ranges or bearings can also produce ambiguous candidate intersections; use a credible DR area and independent information to select the physically consistent position.

How can observations at different times be combined?

Move the earlier position line through the intervening estimated displacement.

Bring the observations to one time

When position lines are observed at different times, they do not automatically describe one aircraft position. If the first observation is at 12:00 and the next at 12:10, advance the earlier line by the estimated ground displacement over those ten minutes before intersecting it with the later line. At 120 kt GS, that displacement is 20 NM along the estimated track.

Simply intersecting the original lines assumes the aircraft did not move. A running fix inherits the earlier observation error as well as uncertainty in intervening track, GS and timing. It can look as precise on paper as a simultaneous fix while carrying significantly greater uncertainty.

A small error triangle is not a guarantee

Three observed bearings may form a triangle rather than a single point. The true position is not guaranteed to lie inside it: a common systematic error can displace all lines in a similar way. Assess the source, reference and geometry of each observation, not just the size of the drawn triangle.

A new fix far outside the credible DR area calls for investigation of both. Possible causes include a wrong station, misidentified feature, incorrect timestamp, wrong north reference or wind estimate. Agreement between two displays may be weak evidence when they use the same underlying navigation source.

Quantify the transport uncertainty

For a ten-minute interval, a 6 kt error in the estimated advancing velocity adds 1 NM of line-transport error. That uncertainty is additional to the original bearing error and can be amplified by the final crossing geometry. A shorter interval can reduce transport uncertainty but may give a smaller change in observation angle, worsening geometry. The best interval is therefore a trade-off, not simply the longest or shortest possible. Plan the observations so that both timing and useful crossing geometry are available.

Can you explain and apply the mechanism?

Four original questions. Work from the stated facts, then compare your reasoning with the explanation.