Nearly parallel lines: geometry amplifies uncertainty

A line constrains a possible position without independently selecting one point. Two non-parallel lines have a mathematical intersection.

That does not mean the point is as well determined as the crisp drawing suggests: small changes in the lines can move the intersection substantially. We explore sensitivity on an abstract grid without coastal targets, instruments or procedures for fixing a position at sea.

In the first model, line A is horizontal and exact: y=0. Line B initially passes through the origin and makes angle θ with A.

We shift B parallel to itself by distance d, measured perpendicular to B. We neither rotate the line nor move A.

These conditions matter: another combination of changes would be another problem. The new crossing remains on A but moves away from the origin.

The relationships form a right triangle. The side of length d is opposite angle θ, while the crossing’s displacement along A is the hypotenuse.

Thus sin(θ)=d/displacement. As θ approaches zero, its sine decreases and the same perpendicular shift corresponds to greater displacement.

The formula quantifies geometric amplification rather than compass accuracy or a confidence level.

If B’s shift can range between −2 and +2 m, the model admits a parallel strip rather than one line. Where the strip crosses A, it produces an interval.

At 30° this is −4 to +4 m; at 10°, approximately −11.518 to +11.518 m. These are results admitted by our invented bounds, not automatically a 95% interval: no probability distribution has been assigned to the shifts.

If A is uncertain too, two strips can intersect in a region whose shape depends on angles and widths. If both share a systematic displacement, the point or region can be displaced together.

Adding lines need not remove that common effect. Having more mathematical relationships and having independent relationships are different properties; information origin remains relevant.

An angular change differs from the parallel translation used in our calculation. Rotating a line around a point changes its separation elsewhere; separation can be larger farther from the rotation centre.

A degree-valued error cannot be inserted in place of d in metres. Another model relating angle, distance and reference would first be needed, with units retained.

Exactly parallel distinct lines do not intersect; coincident lines admit infinitely many points. This limit explains why the formula grows without supplying a position when sin(θ)=0.

A computer can draw an intersection with many decimal places without creating further information about its inputs. Understanding the geometry supports learning about bearings and position relationships; executing and validating real observations remains a separate applied competence.