61 solved problems

Horizontal and vertical curves solved from plan data and taken all the way to the stakes: stations, deflection angles, chords, offsets and grade elevations.

• Stake a simple circular curve by deflection angles from the PCA 1150.00 ft radius curve with a 28°42′00″ central angle is solved for every element, stationed through the PI, and taken to a full deflection-angle and chord table for staking from the PC.20 min
• Recover a curve radius from a measured long chord and middle ordinateAn existing curve with no plan record is measured with a tape: the long chord between two identifiable points and the middle ordinate at its center. Radius and central angle follow in closed form.18 min
• Fit a horizontal curve to a required external distanceAn obstruction near the PI forces the curve to stay at least 65.00 ft out from the intersection point. The minimum radius is solved from the external distance, rounded to a design value, and stationed.18 min
• Degree of curve on plan data: arc definition against chord definitionA plan labels a curve D = 3°30′00″ with Δ = 24°15′00″ but does not say which definition of degree of curve applies. Both radii are computed and the two sets of elements compared.22 min
• Curve stationing and the PT equals PC plus L trapA 900 ft radius curve is stationed from its PI, with the 29.63 ft of chainage lost at the corner made explicit and every full station on the curve tabulated.16 min
• Stake a circular curve by tangent offsets and by chord offsetsA 700 ft radius curve is laid out with a tape alone: first by rectangular offsets from the back tangent, then by successive offsets from the prolonged chord, with the approximate and exact formulas compared.24 min
• Radial staking of curve points by coordinates from a control stationCurve stations are converted to coordinates through the PC and the back tangent, then inversed to a control point to give the angle and distance a total station needs for radial layout.30 min
• Compound curve: tangent distances and stationing from the main PITwo arcs of different radius meet at a point of compound curvature. The vertex triangle gives the two tangent distances back to the main PI, and the alignment is stationed through both arcs.30 min
• Equal-tangent crest vertical curve: grade elevations and the high pointA 600 ft crest curve joining a +3.20 percent grade to a −2.40 percent grade is reduced to a grade sheet at full stations, with the high point located and the K value reported.22 min
• Sag vertical curve: low point, overhead clearance and the drainage checkA 500 ft sag curve is reduced to a grade sheet, the low point is located for drainage, clearance under an overhead structure is verified, and the flat zone around the low point is measured against a 100 ft criterion.28 min
• Vertical curve length fitted to a fixed elevation at a fixed stationA sag curve must pass through an existing manhole rim at a set station and elevation. The required length comes out of a quadratic with two roots, only one of which puts the point on the curve.30 min
• Superelevation runoff and tangent runout layout for a highway curveA 6 percent superelevation transition is laid out around a 1000 ft radius curve: runoff and runout lengths from the relative gradient, the five key stations, and edge-of-pavement elevations for the stakes.32 min

Closed traverses computed from field notes: angular closure, azimuths, latitudes and departures, misclosure and precision, compass-rule adjustment, final coordinates.

• Closed-loop traverse from field angles to compass-rule coordinatesA five-sided loop traverse is taken from raw interior angles and taped distances through angular closure, azimuths, latitudes and departures, misclosure and precision, to compass-rule adjusted coordinates.35 min
• Transit-rule adjustment compared with the compass ruleThe same five-sided loop is readjusted by the transit rule, which distributes the closure in proportion to each latitude and departure rather than to course length, and the two sets of coordinates are compared.25 min
• Closed-link traverse between two pairs of control monumentsA four-course traverse runs from one control pair to another. Azimuth closure comes from the two published control azimuths and position closure from the published coordinates of the far monument.38 min
• A traverse that fails its precision specification, and what to do about itA four-sided traverse closes at 1:2,538 against a 1:10,000 requirement. The direction of the closure line points straight at the course carrying the blunder, which is remeasured and the traverse recomputed.28 min
• Omitted measurement: the missing bearing and distance of one sideFour sides of a five-sided parcel are recovered but the fifth runs through a swamp. Both its bearing and its length are computed from the condition that the traverse must close.22 min
• Traverse with two missing lengths solved from the closure conditionsTwo sides of a five-sided traverse have known directions but no measured lengths. The two closure conditions give a pair of linear equations that solve both lengths at once.26 min
• Angular closure with a suspected blunder in one angleA six-sided traverse misses its angular closure by 2′36″ against a 37 second tolerance. The size of the misclosure identifies a single blunder, which is traced in the field book and removed before any adjustment.30 min
• Closed traverse with the area required as well as the closureA six-sided parcel traverse is closed and adjusted, then its area is computed twice, once by the coordinate method and once by double meridian distances, so the two can be made to agree.34 min
• Closed traverse computed from deflection anglesA five-sided loop observed with deflection angles rather than interior angles. The closure condition becomes 360 degrees rather than (n − 2) times 180, and azimuths advance by simple addition.30 min
• Reducing a ground traverse to grid with a combined factorA link traverse measured on the ground is tied to state plane control. The combined factor is built from the grid scale factor and the elevation factor, and the closure is computed with and without it.32 min
• Locating a new point by intersection from two traverse stationsA point on the far side of a river is fixed by directions observed from two established traverse stations, then verified independently by the sine rule and by a third-angle check.24 min
• Metes and bounds deed traverse: closure, precision and areaA five-call deed description is reduced to a traverse. It closes at 1:4,958, which is examined against what a deed of that vintage should be expected to deliver, then adjusted and its area compared with the recited acreage.26 min

Areas by coordinates and by DMD, curved boundaries, and the partition problems that split a parcel to a required area by a line of given direction or through a given point.

• Area of a closed parcel from coordinates by the shoelace ruleA five-corner parcel is given as plane coordinates. Work the cross-product sum term by term to square feet and acres, and use the sign of the result to prove the corner list is in boundary order.18 min
• The same parcel by double meridian distanceThe HAWTHORN TRACT area recomputed on a DMD sheet — the form a plat's computation panel still shows — and set against the coordinate answer to prove the two methods are the same computation.22 min
• Area of a parcel with one circular-arc boundaryA five-sided parcel whose southerly boundary is a circular arc. Compute the area of the chord polygon by coordinates, add the circular segment R²/2·(Δ − sin Δ), and check the segment as sector minus triangle.22 min
• Area from a metes-and-bounds descriptionA deed of five calls reduced to latitudes and departures, carried to coordinates, and closed out to an area — with the description's own closure tested before the area is trusted.26 min
• Partitioning a parcel by a cut line of given directionThree acres are to be severed from an eight-acre tract by a line parallel to the frontage. Set up the cut area as a quadratic in the perpendicular offset, solve it, and stake the two new corners.32 min
• Partitioning a parcel by a line through a fixed pointA cut line must start at an existing monument on the frontage and sever 2.400 acres. Swing a trial line to a corner, then rotate it off that corner by the triangle the deficiency demands.34 min
• Irregular boundary area by the trapezoidal rule and by Simpson's one-third ruleNine offsets to a meandering creek bank at 25 ft intervals, reduced two ways. The two rules differ by 22.5 ft², and the reason they differ is the reason to prefer one of them.16 min
• Area of a PLSS aliquot part reconciled against a measured traverseThe nominal forty of a quarter-quarter set against the area its aliquot corners actually enclose. Build the aliquot corners by proportion from measured section corners, then compute what the description really conveys.30 min
• Converting a computed area between square feet, acres and hectaresOne parcel area expressed in every unit a plat may call for, and the 4 ppm consequence of the two American feet made explicit — because 2 ppm in a length is 4 ppm in an area.14 min
• Area of a right-of-way strip take along an alignmentA road widening takes a fifteen-foot strip along a frontage that runs partly on tangent and partly around a curve. The tangent part is a rectangle; the curved part is the difference of two circular sectors.20 min
• Parcel area from a compass-rule-adjusted traverseField bearings and distances taken all the way through: closure, compass-rule adjustment, coordinates, and the area those adjusted coordinates enclose — with the adjustment's effect on the acreage measured rather than assumed.38 min

Random error carried through sums, series, products and functions; weights and weighted means; the precision a procedure can actually deliver.

• Standard deviation, standard error and a rejection decision on repeated distancesTen EDM observations of one baseline, reduced to a residuals table, a standard deviation and a standard error of the mean — and one suspect observation tested against a criterion that can actually reject it.22 min
• Error in a taped distance measured in n tape lengths, against a single EDM shotThe same 648 ft line measured two ways. Series propagation over seven tape settings against one EDM observation with a constant-plus-ppm specification, and what the comparison says about when taping is still adequate.16 min
• Error propagated into a computed area from errors in two sidesA rectangular parcel measured on two sides, and the uncertainty those two measurements put into the acreage — with the fractional form of the same result as a check and the worst case as a bound.14 min
• Error propagated into latitude and departure from errors in an angle and a distanceOne course, one distance uncertainty and one azimuth uncertainty, carried through the general propagation formula with partial derivatives — and a rotation argument that checks the answer without repeating it.26 min
• Weighted mean of three observations of differing precisionThree crews, three standard errors, one distance. Weights as the inverse of the variance, the weighted mean, the standard deviation of unit weight, and what it means when that comes out well above one.22 min
• Predicted traverse closure error against the observed misclosureWhat misclosure should a five-course traverse produce, given the instrument that ran it? Propagate the per-course errors, predict the closure, and compare it with what the field actually returned.30 min
• Level-loop misclosure judged against a C√M tolerance and distributedA four-leg differential level loop that fails to return to its benchmark by 0.040 ft. Test it against C√M for two orders of accuracy, distribute the misclosure by leg length, and carry the adjusted elevations round.20 min
• Angular closure of a seven-sided traverse against a k√n toleranceSeven interior angles that sum 21 seconds over the geometric condition. Test the misclosure against the k√n tolerance a 10-second instrument justifies, then distribute it equally and prove the adjusted angles close.18 min
• Combining a systematic correction with a random errorA tape 0.012 ft short and thirteen settings of it. The systematic part is removed by arithmetic and the random part is carried as an uncertainty — and the two must never be combined in quadrature, which is the mistake the problem exists to prevent.20 min

Differential and trigonometric levelling reduced from notes, loop closure against a tolerance, and distribution of the misclosure.

• Differential level circuit reduced from field notesReduce a four-setup differential level run from BM-1 to BM-2 through three turning points, then prove the page with the sum of backsights against the sum of foresights.15 min
• Closed level loop tested against a tolerance and adjustedA four-leg level loop returns to its starting benchmark 0.060 ft out. Test the misclosure against a C times root-M tolerance, then distribute it in proportion to leg length and compute adjusted elevations.20 min
• Level net of three loops adjusted by inspectionThree benchmarks and an interior junction point are connected by six level lines forming three independent loops. Adjust the net by inspection, distributing each loop misclosure by line length and iterating until every loop closes.35 min
• Profile levelling reduced to a profile and a grade lineReduce a profile level run with intermediate foresights to ground elevations at six stations, then compute a grade line at -1.20 percent and the cut or fill at each station.22 min
• Cross sections reduced to end areas and an earthwork volumeTwo roadway cross sections in cut are reduced from slope-stake notes to end areas by the coordinate method, and the volume between them is computed by average end area.30 min
• Trigonometric levelling with the curvature and refraction correctionA single long total-station shot is reduced to an elevation, carrying instrument and target heights and the combined correction for earth curvature and atmospheric refraction.18 min
• Two-peg test worked to a collimation correctionA two-peg test on a level is reduced to a collimation error per unit sight length, expressed in seconds of arc, and then used to correct a pair of unbalanced sights and recover a turning-point elevation.20 min
• Reciprocal levelling across a river crossingA 1,360 ft river crossing is levelled reciprocally from both banks; the mean of the two differences removes the systematic error, and the half-difference is compared with the predicted curvature and refraction.22 min
• Benchmark carried on two routes and reconciled by weighted meanA new benchmark is levelled from two published control points over routes of different length. Test the discrepancy, weight each route inversely with its length, and compute the weighted mean elevation and its standard deviation.18 min

Generic retracement computations: proportionate measurement, aliquot subdivision of a section, and reconciling record and measured dimensions.

• Lost corner restored by single proportionate measurementA section corner on a straight range line is lost between two existent original corners. Restore it by single proportionate measurement, working in coordinates and proving the record ratio is preserved.20 min
• Lost interior section corner by double proportionate measurementAn interior section corner is lost with existent corners a mile or so in each of the four cardinal directions. Restore it by double proportionate measurement and verify that the record latitude and departure ratios are preserved.30 min
• Center of section by intersection of the quarter-corner linesA section is subdivided from eight recovered monuments. The center of section is fixed by intersecting the north-south and east-west quarter-corner lines, and the four quarter-section areas are computed and checked against the whole.28 min
• Areas of government lots on a north township boundaryThe north tier of lots in a fractional section is computed from a retracement of the north boundary, with the deficiency of the mile falling in the last half-mile.22 min
• Record chains reconciled against a measured distanceThree record calls in chains between two existent monuments do not sum to the measured distance. Distribute the difference proportionately and report the restored positions in both feet and chains.15 min
• Laying out the SW¼ of the NE¼ from section corner coordinatesFrom four section corner coordinates, protract the quarter corners, fix the center of the section and the center of the north-east quarter, then lay out the SW¼ of the NE¼ and compute its area.30 min
• Retracing a metes-and-bounds description that will not closeA four-course deed fails to close by more than nine feet. Locate the error from the direction of the misclosure, decide what to hold, correct the record, and adjust the retraced parcel by the compass rule.35 min
• Corner position recovered from two bearing-tree tiesTwo original bearing trees are found and coordinated. The corner is computed independently from each recorded tie, the discrepancy is examined, and the accepted position is checked against the record distance between the trees.25 min