Topografía - Cálculo y compensación de poligonales

Main ideas and lesson conveyed

The video explains a complete office workflow for calculating and adjusting a closed traverse (topographic polygonal) after collecting field data. It corrects two main error types:

1. Angular error (from horizontal angle measurements)
2. Linear error (from distance/side projections)

The objective is to produce more accurate adjusted angles, azimuths, projections, and vertex coordinates, so the traverse closes consistently with theoretical geometry.

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Concepts and methodology (step-by-step)

0) Given / traverse description (context)

• The traverse is closed and has 5 vertices (named in the video as A, B, C, D, and E/I, depending on transcription).
• Orientation
  • Measured counterclockwise starting from North.
  • Because of the measurement direction, internal angles are used for azimuth propagation.
• Field inputs include
  • Horizontal angles at each vertex (internal angles)
  • A starting azimuth for the initial alignment: 113° 13′ 24″
  • Side distances between successive vertices to compute projections

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1) Angular closure error (eₐ)

Purpose: quantify how far the measured internal angles deviate from the theoretical sum.

1. Compute actual sum of internal angles

   • Measured internal angles (5 angles):
     • 86° 56′ 20″
     • 162° 0′ 10″
     • 119° 25′ 14″
     • 74° 49′ 34″
     • 96° 49′ 32″
   • Actual sum: 539° 59′ 50″

2. Compute theoretical sum of internal angles

   • Formula used:
     • theoretical sum = 180 × (n − 2), where n = number of vertices
   • For n = 5:
     • 540° (since 180 × 3 = 540°)

3. Angular closure error

   • eₐ = actual − theoretical
   • eₐ = 539° 59′ 50″ − 540° = −0° 0′ 10″
   • Reported as approximately 10 seconds (negative indicates under/over relative to theory)

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2) Angular tolerance (check before adjustment)

Purpose: determine whether the angular error is acceptable based on instrument resolution.

1. Angular tolerance formula (as stated)

   • TEA = a × √n
   • where:
     • a = instrument resolution (given as about 20″)
     • n = number of vertices (5)

2. Compute tolerance

   • TEA ≈ 20″ × √5 ≈ 44″72 (as stated)

3. Verification

   • Compare |eₐ| to TEA
   • Angular error (~10″) < tolerance (~44″72) → proceed with angular compensation

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3) Angular compensation (distribute the angular error)

Purpose: adjust each measured internal angle so their sum equals the theoretical value.

1. Angular correction per angle

   • cα = eₐ / n
   • With eₐ ≈ −10″ and n = 5:
     • cα ≈ −2″ (the video notes the sign convention for applying correction)

2. Corrected angle table

   • For each vertex angle:
     • corrected angle = original angle + (correction with appropriate sign)
   • The video applies +2″ to each original angle (because the correction uses the opposite sign of the error).

3. Check

   • Sum of corrected angles should equal the theoretical sum (540°).

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4) Azimuth calculation using “law of propagation” (azimuth propagation)

Purpose: compute the azimuths of each alignment sequentially.

Rules presented (as stated)

When measured counterclockwise from North and using internal angles, the video provides propagation logic equivalent to standard azimuth rules:

1. If (angle at vertex + current azimuth) > 180°, subtract 180°.
2. If (angle + previous azimuth) ≤ 180°, add 180°.
3. If the previous azimuth exceeds 540°, subtract 540°.

Procedure

1. Start with known initial azimuth:
   • AZ(A→B) = 113° 13′ 24″
2. For each next alignment:
   • Add the corrected internal angle at the relevant vertex
   • Apply the rule to keep azimuth within the required range
   • Continue until the last alignment is consistent
3. The video demonstrates intermediate azimuth computations, including an example:
   • AZ ≈ 95° 13′ 36″ after applying the “subtract 180°” adjustment
   • Then it similarly derives subsequent azimuths until the final azimuth is obtained

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5) Side projection calculations (ΔN and ΔE)

Purpose: convert each side length and azimuth into northing and easting components.

• Interpretation (Cartesian):
  • East axis = horizontal
  • North axis = vertical
• For a side of length L and azimuth θ, the video uses sine/cosine relationships in the form:
  • ΔN = L × (appropriate trig with θ)
  • ΔE = L × (appropriate trig with θ)
• The video computes ΔN and ΔE for every side and notes signs based on direction/quadrant.

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6) Linear closure error (δ)

Purpose: quantify the mismatch when summing projections around the closed traverse.

1. Sum north projections:
   • ΣΔN → residual around 0.05
2. Sum east projections:
   • ΣΔE → residual around −0.38
3. Compute linear error magnitude:
   • eL = √( (ΣΔN)² + (ΣΔE)² )
   • √(0.05² + (−0.38)²) ≈ 0.06 m
   • Reported linear error: 0.06 meters

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7) Linear tolerance (allowable range)

Purpose: decide whether the traverse accuracy is sufficient without remeasurement.

1. Linear tolerance formula (as stated):
   • TL = k × √(ΣL)
   • Using ΣL = 390.475 (as shown)
2. Computed tolerance:
   • TL ≈ 0.30 m
3. Verification:
   • 0.06 m < 0.30 m → acceptable; proceed with correction

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8) Correction of projections (compass method / equal proportional adjustment)

Purpose: distribute the linear closure error among each side projection proportionally to its length.

Core correction equations (as stated)

For each side i:

1. North-component correction:
   • vΔNᵢ = − (eΔN / ΣL) × Lᵢ
2. East-component correction:
   • vΔEᵢ = − (eΔE / ΣL) × Lᵢ

Procedure described

1. Use the computed residuals:
   • eΔN ≈ 0.045
   • eΔE ≈ −0.038
   • ΣL = 390.475
2. For each side (at vertices A, B, C, D, E/I):
   • Apply the proportional factor using its distance Lᵢ
   • Compute partial corrections for ΔN and ΔE
3. Compute corrected projections:
   • ΔN_corrected = ΔN_original + vΔN
   • ΔE_corrected = ΔE_original + vΔE
4. The video describes performing these computations in an office spreadsheet and summing corrected values.

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9) Compute adjusted coordinates of vertices

Purpose: convert corrected projections into final northing/easting coordinates.

1. Use coordinate propagation:
   • Nᵢ = N_previous + (± ΔN_corrected)
   • Eᵢ = E_previous + (± ΔE_corrected)
   • Sign depends on whether the corrected projection is positive or negative.
2. The video updates coordinates sequentially from one vertex to the next.
3. It concludes that, after adjustment and compensation, the traverse closes correctly.

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Speakers / sources featured

• No specific individual speaker name is provided in the subtitles.
• The video appears to be an instructional lesson (source not identified in the provided text).
• Speakers identified: none (not explicitly named in subtitles).