Traverse Surveys & Computations (Kavanagh)

Definition: Open Traverse

A traverse that does not close geometrically—it begins at one point and ends at another without returning. Used for highways, pipelines, transmission lines. Requires extra verification since position closure cannot be checked. — Kavanagh Ch. 6, p. 157

Definition: Closed Traverse

A traverse that either (1) begins and ends at the same point (loop traverse), or (2) begins and ends at points whose positions were previously determined. Allows geometric verification of angles and position closure. — Kavanagh Ch. 6, p. 159

Definition: Loop Traverse

A closed traverse that begins and ends at the same point, forming a closed geometric figure. Allows checking both angular closure and position closure. — Kavanagh Ch. 6, p. 159

Interior Angle Sum Formula

Sum of interior angles = (n - 2) × 180° Where n = number of sides Examples: • 4 sides: (4-2)×180° = 360° • 5 sides: (5-2)×180° = 540° • 6 sides: (6-2)×180° = 720° — Kavanagh Ch. 6, p. 160

Methods to Balance Field Angles

1. Equally balanced: Distribute error evenly to each angle 2. Arbitrarily balanced: Adjust specific angles (e.g., if one setup was suspect) Angles MUST be balanced before bearing/azimuth computations can begin. — Kavanagh Ch. 6, p. 160

Definition: Bearing

The direction of a line given by the acute angle (0° to 90°) between that line and a meridian, accompanied by quadrant letters (NE, SE, SW, NW). Example: N 71°11' E — Kavanagh Ch. 6, p. 162

How to Reverse a Bearing

Simply switch the direction letters: • N ↔ S • E ↔ W The bearing angle stays the same. Example: N 71°11' E → S 71°11' W S 44°18' W → N 44°18' E — Kavanagh Ch. 6, p. 163

Bearing Computation Pattern (Clockwise)

When computing bearings clockwise around a traverse: 1. Draw the two courses and a meridian at the station 2. Place known bearing angle and interior angle on sketch 3. Solve for unknown bearing angle The sketch shows whether to add or subtract. — Kavanagh Ch. 6, p. 163-165

Definition: Azimuth

The direction of a line given by an angle measured clockwise from the north end of a meridian. Ranges from 0° to 360°. Note: Some geodetic projects measure from south. — Kavanagh Ch. 6, p. 165

How to Reverse an Azimuth

Add or subtract 180°: • If azimuth < 180°: add 180° • If azimuth ≥ 180°: subtract 180° Example: 71°11' → 251°11' 224°18' → 44°18' — Kavanagh Ch. 6, p. 165

Azimuth Computation (Clockwise)

When working clockwise: New azimuth = Back azimuth − Interior angle When working counterclockwise: New azimuth = Back azimuth + Interior angle — Kavanagh Ch. 6, p. 167

Bearing to Azimuth Conversions

NE quadrant: Az = bearing angle SE quadrant: Az = 180° − bearing angle SW quadrant: Az = 180° + bearing angle NW quadrant: Az = 360° − bearing angle — Kavanagh Ch. 6, p. 170

Definition: Latitude (in traverse)

The north/south rectangular component of a line (ΔN). Latitude = Distance × cos(bearing) Latitude = Distance × cos(azimuth) North = positive (+) South = negative (−) — Kavanagh Ch. 6, p. 169

Definition: Departure (in traverse)

The east/west rectangular component of a line (ΔE). Departure = Distance × sin(bearing) Departure = Distance × sin(azimuth) East = positive (+) West = negative (−) — Kavanagh Ch. 6, p. 169

Closure Check for Latitudes and Departures

For a perfectly measured closed traverse: Σ latitudes = 0 Σ departures = 0 (Plus latitudes equal minus latitudes; Plus departures equal minus departures) — Kavanagh Ch. 6, p. 171

Linear Error of Closure Formula

E = √(Σlat² + Σdep²) Where: • Σlat = sum of latitude errors • Σdep = sum of departure errors • E = linear error of closure — Kavanagh Ch. 6, p. 172

Precision Ratio Formula

Precision Ratio = E / P Where: • E = linear error of closure • P = perimeter of traverse Expressed as 1/n, rounded to nearest 100. Example: 0.17/559.36 = 1/3,290 ≈ 1/3,300 — Kavanagh Ch. 6, p. 173

Common Precision Specifications

• Property surveys: 1/5,000 to 1/7,500 • High-cost urban: 1/10,000+ • Engineering/construction: 1/3,000 to 1/10,000 • Ditched highway: 1/3,000 • Monorail transit: 1/7,500 to 1/10,000 — Kavanagh Ch. 6, p. 174

Compass Rule Formulas

Correction in latitude: C_lat = Σ_lat × (course distance / perimeter) Correction in departure: C_dep = Σ_dep × (course distance / perimeter) Corrections are OPPOSITE in sign to errors. — Kavanagh Ch. 6, p. 177

Steps to Apply Compass Rule

1. Calculate corrections for each course (proportional to distance) 2. Determine sign (opposite to error) 3. Add corrections to original lat/dep 4. Verify: Σ balanced lat = 0, Σ balanced dep = 0 — Kavanagh Ch. 6, p. 177-178

Computing Station Coordinates

Northing_B = Northing_A + Balanced_Latitude_AB Easting_B = Easting_A + Balanced_Departure_AB Start with assumed or known coordinates. Check: Final computation returns to starting coordinates. — Kavanagh Ch. 6, p. 181-182

Why Assume Starting Coordinates?

Values like 1000.00 N, 1000.00 E are chosen to ensure all calculated coordinates remain positive (all stations in NE quadrant). This prevents negative coordinates which can cause confusion. — Kavanagh Ch. 6, p. 181

Angular Error Limits by Precision

Linear Accuracy → Max Angular Error • 1/1,000 → 0°03'26" • 1/3,000 → 0°01'09" • 1/5,000 → 0°00'41" • 1/7,500 → 0°00'28" • 1/10,000 → 0°00'21" Total allowable = E × √n angles — Kavanagh Ch. 6, Table 6.3, p. 176