discuss with respect to positional errors in fixes the theory of the cocked hat and the three bearing formula and their applications,

Of course. This is an excellent topic that gets to the heart of practical navigation and the management of uncertainty. Let’s discuss the theory and applications of the cocked hat and the three-bearing formula with respect to positional errors in fixes.

Introduction: The Reality of Positional Errors

In a perfect world, a navigator would take bearings on three objects, and the resulting Lines of Position (LOPs) would intersect at a single, perfect point on the chart—the vessel’s fix. In reality, this rarely happens. Every observation and every piece of equipment has inherent errors. These errors mean the LOPs will almost always form a small triangle, not a single point. Understanding how to interpret this triangle and how to verify the integrity of your data is a fundamental skill of a navigator. The “cocked hat” and the “three-bearing formula” are two key tools for this purpose.

1. The Cocked Hat

A cocked hat is the triangle formed on a navigation chart when three Lines of Position (LOPs) fail to intersect at a single point.

Theory: Why Does a Cocked Hat Form?

The size and shape of the cocked hat are direct indicators of the magnitude and nature of the errors in your fix. The sources of these errors can be categorized as:

1. Observation Errors:

   • Human Error: Incorrectly reading the compass or pelorus.
   • Timing Error: Especially in a “running fix” where the vessel moves between observations. An error in the time, speed, or course used for the run will displace one or more LOPs.
   • Object Misidentification: Taking a bearing on the wrong landmark or the wrong part of a landmark (e.g., the water’s edge of a point instead of its charted position).

2. Equipment Errors:

   • Compass Error: An unknown or incorrectly applied deviation or variation. This is a constant error that will affect all bearings taken at that time in the same way.
   • Instrument Error: A misaligned pelorus or a faulty bearing compass.

3. Plotting Errors:

   • Inaccurate use of the parallel ruler or protractor.
   • A thick pencil line, which introduces ambiguity.

Application: Interpreting and Using the Cocked Hat

A navigator doesn’t just discard a fix that results in a cocked hat. They analyze it.

a) Small Cocked Hat: If the triangle is small and the bearings were taken from well-spaced objects (ideally 60° to 120° apart), it indicates a high-quality fix with minimal error.

• Application: For practical purposes, the vessel’s position is assumed to be at the center of the triangle.

b) Large Cocked Hat: A large triangle is a red flag. It signifies a significant error in at least one of the bearings or in the plotting process.

• Application:
  1. Do not trust the fix. Do not assume your position is in the center.
  2. Troubleshoot: The navigator must immediately re-check everything:
     • Were the objects correctly identified?
     • Was the compass error calculated and applied correctly?
     • Were the bearings plotted accurately?
     • Are the compass and pelorus in good working order?
  3. Take a new set of bearings. This is often the quickest way to resolve the issue.

c) Interpreting the Likely Position (The Rules of Thumb):

The shape and orientation of the cocked hat can provide clues about the source of the error.

• Case 1: Constant Error (e.g., Unaccounted Compass Error) If a single, constant error (like a 2° easterly compass error) is applied to all three bearings, it will shift all LOPs parallel to themselves. The result is that the vessel’s true position is outside the cocked hat.
  • The Rule: The true position lies on the side of the cocked hat corresponding to the LOP from the middle object (the one whose bearing is between the other two). Specifically, it will be in one of the three sectors outside the triangle. To be safe, the navigator should assume the position is at the vertex of the cocked hat that places the vessel in the greatest danger (e.g., closest to a shoal or rocks).