9.1 Azimuths, Bearings, Backbearings, and Deflections

9.1 Azimuths, Bearings, Backbearings, and Deflections

Quick Answer: In US civil practice, a north azimuth is the clockwise angle from north, running 0° to 360°. A quadrant bearing is the acute angle from north or south toward east or west (N/S θ E/W). They name the same line. Back azimuth = azimuth ± 180° (fold the result into 0°–360°). A deflection is turned right or left from the prolongation of the previous tangent, not from north.

This section is the office half of CES Domain III item D: calculating horizontal angles as azimuths, bearings, backbearings, and deflections. Chapter 6 covered occupying, backsighting, and turning. Here you convert the bookkeeping so latitudes, departures, and route stationing all use one consistent direction.

The North Convention Used Here

North azimuth, clockwise, 0°–360°. That is ordinary US highway, municipal, and civil-engineering practice, including California engineering-surveying notes on assumed systems and on California Coordinate System of 1983 (CCS83) grid north.

• 0° = north
• 90° = east
• 180° = south
• 270° = west
• 360° = north again (do not publish 360°; write 0°00′)

This is not a south-azimuth system (some older railroad and mining notes), not a counterclockwise mathematical polar angle from the +x axis, and not an interior angle. If a California civil problem says “azimuth” without a qualifier, treat it as a north azimuth.

True north is the geographic meridian. Grid north is the mapped northing axis (CCS83 grid meridians are parallel to the zone’s central meridian, so they match true north only near that meridian). Magnetic north is where a compass points. The Civil Engineering Surveying (CES) exam is not a Professional Land Surveyor (PLS) astronomy exam: you will not compute Laplace corrections or Polaris azimuths. You do need to name which north a given azimuth uses, and you need one short magnetic-declination conversion.

Quadrant Bearings

A bearing in civil notes is a quadrant bearing:

1. Face north or south (whichever meridian end is nearer).
2. Turn an angle θ ≤ 90° toward east or west.
3. Write N θ E, S θ E, S θ W, or N θ W.

Bearings never carry 120°. If you compute 120°, you chose the wrong meridian end. Azimuths never carry a quadrant letter.

Worked Conversions — All Four Quadrants

Bearing to azimuth

• N 18°42′ E (NE): azimuth = 18°42′.
• S 41°28′ E (SE): azimuth = 180°00′ − 41°28′ = 138°32′.
• S 63°15′ W (SW): azimuth = 180°00′ + 63°15′ = 243°15′.
• N 27°50′ W (NW): azimuth = 360°00′ − 27°50′ = 332°10′.

Check the NW line: 332°10′ + 27°50′ = 360°00′.

Azimuth to bearing

Bin the azimuth into a 90° quadrant, then peel θ off north or south — never off east or west.

• Azimuth 74°22′ (0°–90°, NE): bearing N 74°22′ E.
• Azimuth 163°08′ (90°–180°, SE): θ = 180°00′ − 163°08′ = 16°52′, bearing S 16°52′ E.
• Azimuth 218°44′ (180°–270°, SW): θ = 218°44′ − 180°00′ = 38°44′, bearing S 38°44′ W.
• Azimuth 247°35′ (still SW): θ = 247°35′ − 180°00′ = 67°35′, bearing S 67°35′ W.
• Azimuth 304°18′ (270°–360°, NW): θ = 360°00′ − 304°18′ = 55°42′, bearing N 55°42′ W.

The SW conversion is the one people reverse. Subtracting 218°44′ from 270° gives 51°16′ and a west-based label. Civil bearings are not written that way. Always peel SW azimuths from south (180°).

Back Azimuth and Backbearing

The reverse of a line is 180° from the forward direction.

Back azimuth = azimuth ± 180°, then fold into 0°–360°:

• If the forward azimuth is less than 180°, add 180°.
• If the forward azimuth is 180° or more, subtract 180°.

Worked numbers:

• Forward azimuth AB = 74°22′ → back azimuth BA = 74°22′ + 180°00′ = 254°22′.
• Forward azimuth CD = 291°08′ → back azimuth DC = 291°08′ − 180°00′ = 111°08′.

Backbearing is the reverse quadrant bearing: flip both letters, keep θ.

• Forward N 18°42′ E → back S 18°42′ W (azimuth 18°42′ + 180° = 198°42′, which is S 18°42′ W).
• Forward S 41°28′ E → back N 41°28′ W.
• Forward S 38°44′ W → back N 38°44′ E.
• Forward N 55°42′ W → back S 55°42′ E.

Occupying B and backsighting A, the instrument looks along azimuth BA, not AB. Using the forward azimuth as the backsight orientation is a 180° bust in coordinates — the same class of error as the polar-layout trap in Chapter 6.

Deflection Angles from a Tangent

Route and highway notes often book deflection angles instead of azimuths at every PI.

A deflection is the horizontal angle from the prolongation of the previous course (the tangent you were already on) to the next course, labeled right (R) or left (L) as you look forward along the alignment.

• Deflection right increases a north azimuth.
• Deflection left decreases a north azimuth (add 360° if the result crosses 0°).

Outgoing azimuth = incoming azimuth + (right deflection) − (left deflection), then fold to 0°–360°.

Worked route example

Incoming azimuth of the tangent is 112°30′.

• Deflect 18°45′ R: outgoing azimuth = 112°30′ + 18°45′ = 131°15′.
• Same incoming, 18°45′ L: outgoing = 112°30′ − 18°45′ = 93°45′.

Those two next courses are 37°30′ apart (twice the deflection). Right versus left is a different line, not a rounding issue.

Chain two deflections on a simple alignment:

• Start azimuth 83°10′.
• Deflect 9°36′ R → 83°10′ + 9°36′ = 92°46′.
• Then deflect 22°18′ L → 92°46′ − 22°18′ = 70°28′.

Circular-curve layout uses a related idea (the deflection from the tangent at the PC to the PT is Δ/2). That geometry is Chapter 10. In this section, treat a deflection as a route-book angle from a tangent.

Magnetic, True, and Grid — Short Version

Three “norths” appear on California civil jobs:

Magnetic declination is the angle from true north to magnetic north, labeled east or west. In much of California the needle currently points east of true north, but you use the value printed on the source, not a memorized statewide constant.

Mnemonic for azimuths: declination east, magnetic least — a magnetic azimuth is smaller than the true azimuth when declination is east.

True azimuth = magnetic azimuth + (east declination)
True azimuth = magnetic azimuth − (west declination)

Worked: magnetic azimuth 48°20′, declination 13°15′ East.

True azimuth = 48°20′ + 13°15′ = 61°35′.

Grid versus true is convergence (the mapping angle). On a small assumed site you often hold project north and never convert (Chapter 5). On a mapped CCS83 job, published azimuths are usually grid. Do not mix a compass azimuth, a true azimuth, and a grid azimuth in one traverse without converting. That is the whole CES-level lesson; leave geodetic astronomy to the PLS exam.

Exam Traps

1. South-azimuth habit — subtracting from 180° when the problem is a north azimuth already in the NE quadrant.
2. SW peel from 270° — azimuth 218°44′ is S 38°44′ W, not a west-based bearing.
3. Backsight uses forward azimuth — occupy B, backsight A: use azimuth BA.
4. Deflection from north — a deflection is from the prolongation, not from north. Adding 18°45′ to 0° instead of to 112°30′ is the wrong line.
5. Compass azimuth treated as grid — convert by declination (and, if the job requires it, convergence) before you compute latitudes.