Converting references without mixing signs

A course is a direction expressed from a reference. Here, angles run clockwise from north and use three digits: 005°, 080° or 275°.

The same physical line can have different values when measured from different north references. Converting a course changes the measurement reference; it does not automatically turn the boat or remove current.

Let C be the compass value, M the magnetic value and T the true value. Deviation d and variation v are positive east and negative west.

Under this convention, M = C + d and T = M + v. Reversing the chain subtracts the same components.

The relationship must be applied to compatible quantities: correcting a compass heading does not turn it into course over the ground.

A circle does not continue beyond 360°. A total of 363° is represented as 003°; −4° becomes 356°.

This is circular arithmetic. Prematurely rounding every component can change the result.

Written precision should remain proportionate to the source information: extra decimal places produced by calculation do not improve the underlying data.

A source may use different abbreviations or present a combined compass error. Before transferring a formula, identify what each symbol means and its sign convention.

These calculations are angular theory without an actual route, position or deviation table. Navigation application needs information valid for the context and an understanding of its use, not merely arithmetic that balances.

Compare two people measuring one line from different zero directions. If one reference is rotated 7° east, measuring from it can give a value 7° smaller than measuring from true north.

Conversion brings both into the same reference. This geometrical reasoning checks calculation direction without relying on a memorised “add east” phrase.

Each value still needs its symbol: 085° should state whether it is T, M or C.

A reciprocal direction points oppositely along the same line. Adding 180° and normalising gives the reciprocal: 085° T becomes 265° T, while 275° T becomes 095° T.

The reference is retained. “Reciprocal” differs from “convert to magnetic”: the former reverses direction, the latter changes measurement zero.

Distinguishing them avoids misorienting an observation line by mixing two operations.