6.2 Measuring Angles and Establishing Points
Key Takeaways
- A horizontal angle is interior (inside a figure), a deflection (left or right from the prolongation of the previous course), or the change in azimuth (clockwise from north, 0° to 360°).
- Direct and reverse (plunge, face left/face right) is a field procedure: average the two pointings to cancel collimation and similar instrument errors. Chapter 9 treats the error budget; the field habit is to mean D and R.
- Polar layout is unique: occupy a known point, backsight a second known point, turn an angle, measure HD. Distance-distance intersection of two circles has two solutions unless you know which side of the baseline.
- Worked polar: A (N = 5,000.00, E = 2,000.00), B (N = 5,300.00, E = 2,400.00); occupy B, backsight A, turn 90° right, HD = 120.00 ft → C at N = 5,396.00 ft, E = 2,328.00 ft.
- A line is defined by two points. Intermediate points are ranged in; the line is extended by occupying the forward point, backsighting the rear point, and plunging. Backsight orients; foresight sets the new point.
6.2 Measuring Angles and Establishing Points
Quick Answer: To locate a point you combine a horizontal angle with a horizontal distance. Orient by backsighting a known point, turn to the new direction, foresight the target, and measure HD. Direct and reverse (plunge) averages face-left and face-right to cancel instrument pointing errors. Polar (angle + distance) is unique; distance-distance generally has two intersections.
Horizontal Angles: Interior, Deflection, and Azimuth
Three common ways to book the same field pointing:
Interior angle — the angle inside a polygon at the occupied station, between the backsight and the foresight. On a closed traverse the interior angles have a geometric sum you will use in Chapter 8. In the field you still occupy, backsight, and turn; “interior” is how you label the result.
Deflection angle — the angle from the prolongation of the previous course to the next course, booked right (R) or left (L). Route work and many construction alignments use deflections. If the incoming azimuth is 75°00′ and you deflect 12°00′ R, the outgoing azimuth is 87°00′. A left deflection subtracts (and you add 360° if the result goes negative).
Azimuth — the clockwise angle from north to the course, 0° to 360°. Civil field notes on assumed systems often use an assumed north; published work may use grid north. Convert bearing to azimuth before computing latitudes and departures: N 36°20′ E → azimuth 36°20′; S 42°10′ E → azimuth 180° − 42°10′ = 137°50′; S 10°00′ W → 190°00′; N 10°00′ W → 350°00′.
Back azimuth (backbearing as a 180° reverse) = azimuth ± 180° (keep the result in 0°–360°). Occupying B and backsighting A, the instrument is looking along azimuth BA, not AB.
| Recorded quantity | Zero / reference | How you use it in the field |
|---|---|---|
| Interior angle | Backsight line at the occupied station | Traverse angles; polygon closure |
| Deflection R/L | Prolongation of the previous course | Route alignment, curve deflection layout |
| Azimuth | North, clockwise 0°–360° | Coordinates: ΔN = HD cos(az), ΔE = HD sin(az) |
Coordinate reductions (this chapter’s field step, expanded in Chapter 8):
ΔN (latitude) = HD × cos(azimuth)
ΔE (departure) = HD × sin(azimuth)
Use HD, not SD. Use azimuth, not an interior angle, in those two formulas — convert first.
Direct and Reverse (Plunge) at Field-Procedure Level
Direct (face left): telescope in the ordinary position. Reverse (face right): plunge (rotate the telescope 180° about the horizontal axis) and rotate the alidade 180° about the vertical axis so you point the same line with the opposite face.
Why bother: horizontal collimation (line of sight not perpendicular to the vertical axis) and some vertical-axis / plate-bubble effects reverse sign when you change face. The mean of direct and reverse cancels those pointing errors for the horizontal angle. Chapter 9 names the error sources; the field rule is simpler:
- Point the backsight direct, point the foresight direct, record the angle (or directions).
- Plunge, point the backsight reverse, point the foresight reverse, record.
- Mean the two angles. Large D−R disagreement is a pointing blunder or a loose instrument, not something to “adjust later.”
Example: direct angle 47°12′30″, reverse 47°12′50″. Mean = 47°12′40″. The 20″ split is information; the value you carry forward is the mean. Direct-only topography sideshots are common; control angles and layout that must match coordinates should see both faces when the specification cares about tens of seconds.
Backsight and Foresight
Backsight (BS) — the sight that orients the instrument to a known direction (a known point, or a known azimuth).
Foresight (FS) — the sight to the new point you are locating or setting.
(The word “foreshortening” is a drawing effect, not a field sight. Leveling uses BS/FS for rods; angle work uses the same names for the known and unknown directions.)
A setup that never backsights a known point is not oriented. A beautiful HD to a hub with a guessed zero on the plate does not produce coordinates.
Locating a Point: Polar, Intersection, Distance-Distance
Angle-distance (polar, radiation, side shot) — occupy known A, backsight known B, turn a horizontal angle to C, measure HD to C. One solution. This is how most topographic sideshots and most construction points are set.
Angle-angle (intersection) — occupy A, sight C (or turn the angle from AB to AC); occupy B, sight C. Point C is the intersection of two directions. Unique if A and B are distinct and C is not on line AB. Useful when you cannot measure distance (across a channel) but can occupy two control points.
Distance-distance — measure HD from A to C and from B to C. Two circles intersect in two points, symmetric across AB, unless C is on AB (tangent / degenerate). You must know which side of the baseline (a sketch, a right/left call, a third check distance).
| Method | What you measure | How many locations |
|---|---|---|
| Polar (angle + HD) | One angle from a known line, one HD | One |
| Angle-angle | Direction from A and from B | One (off the baseline) |
| Distance-distance | HD from A and from B | Two; pick the correct side of AB |
Establishing a Line: Two Points, Ranging, Extending
Two points define a unique straight line. Ranging places intermediate points on that line: range poles lined in by eye from one end, then checked with the instrument (angle 180° / 0° deflection). Construction centerline hubs between two terminals are ranged and then measured by HD along the line.
Extending a line past the forward point: occupy the forward point, backsight the rear point, plunge the telescope. The line of sight is now the prolongation. For a tighter extension, double-center: plunge and set a temporary point; rotate 180°, backsight again, plunge again; split any difference. That is the same face-change idea as direct/reverse, applied to a 180° line.
Do not “extend” by occupying a random nearby hub and guessing a parallel. Parallels require a computed offset, not an eyeball.
Worked Example — Polar Coordinates from Two Control Points
Control:
- Point A: N = 5,000.00 ft, E = 2,000.00 ft
- Point B: N = 5,300.00 ft, E = 2,400.00 ft
ΔN_AB = 300.00 ft, ΔE_AB = 400.00 ft. Length AB = √(300² + 400²) = 500.00 ft (a 3-4-5 triangle × 100).
Azimuth AB = arctan(400/300) = 53.130102° (53°07′48″).
Occupy B, backsight A, turn a horizontal angle of 90°00′00″ to the right (clockwise), measure HD = 120.00 ft to new point C (prism plumbed; HD already reduced).
Azimuth BA = 53.130102° + 180° = 233.130102°.
Turn 90° right: azimuth BC = 233.130102° + 90° = 323.130102°.
cos(323.130102°) = +0.800000, sin(323.130102°) = −0.600000.
ΔN_BC = 120.00 × 0.800000 = +96.00 ft
ΔE_BC = 120.00 × (−0.600000) = −72.00 ft
C: N = 5,300.00 + 96.00 = 5,396.00 ft
C: E = 2,400.00 − 72.00 = 2,328.00 ft
Trap: adding 90° to azimuth AB instead of BA sends you the wrong way. Azimuth AB + 90° = 143.130102°; then ΔN = 120 × (−0.800000) = −96.00 and ΔE = 120 × 0.600000 = +72.00, giving N = 5,204.00, E = 2,472.00 — which is √(192.00² + 144.00²) = 240 ft from the correct C. Always ask: which direction was the backsight?
Deflection check on a simple tangent
Incoming azimuth 75°00′, deflection 12°00′ R, HD = 80.00 ft. Outgoing azimuth = 87°00′.
ΔN = 80.00 × cos(87°) = 4.19 ft
ΔE = 80.00 × sin(87°) = 79.89 ft
If that deflection had been booked left, azimuth would be 63°00′ and the easting would shrink: ΔE = 80 × sin(63°) = 71.28 ft. Right versus left is a full coordinate miss, not a rounding issue.
Angle-angle sketch numbers
A and B are 400.00 ft apart. Angle at A toward C is 40°, angle at B toward C is 60°, so angle at C is 80°. Law of sines: AC / sin(60°) = 400.00 / sin(80°). sin(80°) = 0.984808, sin(60°) = 0.866025. AC = 400.00 × 0.866025 / 0.984808 = 351.75 ft, BC = 400.00 × sin(40°) / sin(80°) = 400.00 × 0.642788 / 0.984808 = 261.08 ft. Those distances are what an EDM should read if you later occupy A or B and shoot C — an independent check on an intersection.
Field Sequence Worth Memorizing
- Recover two control points; confirm HD between them (here 500.00 ft on a 3-4-5).
- Occupy, level up, backsight, set a circle (zero on the backsight, or a known azimuth).
- Turn the design angle direct; plunge and turn reverse; mean if it is a control or layout angle.
- Measure HD (not SD) to the new point; plumbing the prism or rod is part of the distance.
- Compute N, E (or station/offset) before you leave. A 90° turn on a 3-4-5 pair should look like the 0.8/0.6 split above — if it does not, the azimuth of the backsight is wrong.
Exam Traps
- Interior angle stuffed into the cosine formula — convert to azimuth first.
- Backsight azimuth forgotten — turning 90° from AB rather than from BA.
- Distance-distance treated as unique — two intersections.
- Plunge skipped on control angles — direct-only collimation left in the layout.
- SD used as HD in the coordinate step (Section 6.1 all over again).
Point A is N = 5,000.00 ft, E = 2,000.00 ft and point B is N = 5,300.00 ft, E = 2,400.00 ft. Occupying B, backsighting A, turning 90°00′ to the right, and measuring HD = 120.00 ft, the coordinates of C are which of the following?
In the field, what is the purpose of measuring a horizontal angle direct and reverse (plunging, both faces) and using the mean?
A point C is to be located by measuring only the horizontal distances from two known points A and B (distance-distance). Which statement is correct?