Reading bearings, distances and curves in a legal description
Most errors in legal descriptions are errors in how a call is written. Knowing exactly what each part of a call means makes them easier to spot.
Quadrant bearings
A quadrant bearing gives a direction as an angle of 0° to 90° measured from north or south toward east or west, such as N 45°30'15" E. The first letter says which end of the meridian you measure from, and the last letter says which way you turn.
The same bearing is written many ways in real documents, including North 45 degrees 30 minutes 15 seconds East, N45-30-15E, and N 45°30'15" E. A call that runs exactly along a meridian or parallel is often written simply as North, South, East, or West.
Converting to azimuth
An azimuth is the angle clockwise from north, from 0° to 360°. Calculations use azimuths because they need no quadrant letters.
| Bearing | Azimuth |
|---|---|
| N θ E | θ |
| S θ E | 180° − θ |
| S θ W | 180° + θ |
| N θ W | 360° − θ |
The reverse of a bearing swaps both letters and keeps the angle. The reverse of N 45°30'15" E is S 45°30'15" W.
Curve elements
A curve call describes a circular arc. The common elements are:
- Radius (R): the distance from the curve to its center.
- Delta (Δ): the central angle the arc turns through.
- Arc length (L): the distance along the curve.
- Chord (C) and chord bearing: the straight line from the start of the curve to its end.
- Direction: whether the curve turns to the left or the right as you travel along it.
They are tied together by a few formulas, with Δ in radians for the first:
- L = R × Δ
- C = 2R × sin(Δ ÷ 2)
- Tangent = R × tan(Δ ÷ 2)
Any two of radius, arc, delta, and chord fix the other two. Descriptions often give three or four, which is a useful check: if the given values disagree by more than rounding, one of them is probably mistyped. The checker solves the curve from the pair that agrees best with everything else given and warns about the value that doesn't fit.
Chord bearing and the tangent bearings
The chord bearing sits halfway between the bearing the curve starts on and the bearing it ends on. For a curve to the right, the ending tangent is the starting tangent turned Δ clockwise, and the chord is turned Δ ÷ 2. For a curve to the left, both turn counterclockwise. Getting the direction wrong leaves the end point in the wrong place, so it is one of the first things to check when a description with curves fails to close.