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Omitted measurement cases, traverse plotting

Short answer

Omitted measurements in a closed traverse are solved using ΣLat=0, ΣDep=0. Cases: missing length or bearing of one line, length and bearing of one line, lengths of two lines (sine rule), bearings of two lines (cosine rule), or one length and one bearing (SSA). Plotting by coordinates: compute station coordinates, draw grid, plot stations by (E,N), and join; errors don't accumulate.

Explain different cases of omitted measurements in theodolite traversing. Explain plotting of traverse in grid sheet by co-ordinates method.

Cases of omitted measurement

The sum of latitudes and the sum of departures of a closed traverse are both zero. These two equations allow two unknowns (lengths or bearings) to be found. The known lines are combined into a single resultant with latitude ΣLat\Sigma L_{at} and departure ΣDep\Sigma D_{ep}; the closing line is equal and opposite to it, with length ΣLat2+ΣDep2\sqrt{\Sigma L_{at}^2+\Sigma D_{ep}^2} and bearing given by tan⁡θ=(−ΣDep)/(−ΣLat)\tan\theta=(-\Sigma D_{ep})/(-\Sigma L_{at}), with the quadrant taken from the signs.

CaseUnknown quantitiesMethod
1Length or bearing of one lineDirect from ΣLat\Sigma L_{at}, ΣDep\Sigma D_{ep}
2Length and bearing of one lineClosing line gives both directly
3Lengths of two linesSine rule in the closing triangle
4Bearings of two linesCosine rule in the closing triangle
5Length of one line and bearing of anotherSolve the closing triangle (SSA, two solutions possible)

The affected lines may be adjacent or non-adjacent. If they are not adjacent, the known lines between them are shifted (a closed traverse still closes if the order of its lines is changed), which brings the unknown lines together so that the same triangle method applies.

Plotting by the coordinates method (outline)

  1. Compute the independent coordinates of all stations (from the adjusted latitudes and departures).
  2. Choose a scale, draw a grid sheet and label the grid lines with northings and eastings.
  3. Mark each station at its (E,N)(E,N) position measured from the grid lines, join the stations in order and check that the plotted lengths agree with the field lengths.
  4. Plot details from the stations and complete the map with the title, scale and north arrow.

Plotting by coordinates is preferred to protractor plotting because errors do not accumulate.

✓ Answer checked by Claude Opus 5.5 (high effort)

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