CogoKit

Curve staking

Radial staking of curve points by coordinates from a control station

Curve 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.

Analyze· about 30 minutes by hand· 5 steps

Given

  • A simple circular curve to the right with R = 1000.00 ft and Δ = 32°00′00″.
  • Station of the PC = 26+65.40, with coordinates N 5412.66, E 7338.02.
  • Azimuth of the back tangent at the PC = 74°12′30″, running ahead toward the PI.
  • Control station CP-9 at N 5150.00, E 7600.00, with a reference mark CP-8 at N 5050.00, E 7180.00.
  • Stakes are wanted at every full station and at the PT, set radially from CP-9 with a backsight on CP-8.
  • Coordinates are in US survey feet; azimuths are clockwise from north.

Required

  • The station of the PT.
  • Coordinates of every full station on the curve and of the PT.
  • The distance and the angle right from CP-8 that will set each of those points from CP-9.

Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.

Worked solution

Curve elements and the PT station

The elements come from R and Δ in the usual way. Only the arc length is needed to station the curve, but the long chord is worth having because it will serve later as a check on the coordinate computation.

The PT station is the PC station plus the arc length.

Δ/2 = 16°00′00″ T = 1000.00 × 0.286745 = 286.75 ft L = 1000.00 × 0.558505 = 558.51 ft C = 2000.00 × 0.275637 = 551.27 ft PT = 26+65.40 + 558.51 = 32+23.91

Coordinates of the curve points from the PC

Every point on the curve is reached from the PC by one chord at one azimuth. The chord from the PC to a point at arc distance l is c = 2R sin(l/2R), and the azimuth of that chord is the back-tangent azimuth plus the deflection l/(2R) for a curve to the right, or minus it for a curve to the left.

This is the whole of coordinate curve layout. Nothing accumulates: each point is computed straight from the PC, so a mistake on one row cannot corrupt the next.

Coordinates then follow from the standard forward computation, northing first: N = N(PC) + c cos(azimuth), E = E(PC) + c sin(azimuth).

δ = l / 2000.00 radians; deflection for 100 ft of arc = 2°51′53″ c = 2000.00 sin δ azimuth of chord = 74°12′30″ + δ (curve to the right) N = 5412.66 + c cos(azimuth), E = 7338.02 + c sin(azimuth)

Coordinate table

The first sub-arc is 34.60 ft, from the PC at 26+65.40 to station 27+00.00. The deflection accumulates to exactly Δ/2 = 16°00′00″ at the PT, and the chord there is the long chord.

Note how slowly the northings change past station 30+00 and then reverse: the chord azimuth passes through 90° between 31+00 and 32+00, which is the point where the alignment is running due east and the curve has reached its most northerly ground.

Curve points computed from the PC by chord and azimuth
StationArc from PC (ft)Deflection δChord from PC (ft)Chord azimuthNorthing (ft)Easting (ft)
27+00.0034.600°59′28″34.6075°11′58″5421.507371.47
28+00.00134.603°51′22″134.5078°03′52″5440.487469.61
29+00.00234.606°43′15″234.0680°55′45″5449.567569.16
30+00.00334.609°35′08″333.0483°47′38″5448.667669.11
31+00.00434.6012°27′01″431.1986°39′31″5437.797768.47
32+00.00534.6015°18′55″528.2689°31′25″5417.057866.26
32+23.91 (PT)558.5116°00′00″551.2790°12′30″5410.667889.29

Orient the instrument at CP-9

Radial staking needs one reference direction. Inverse from CP-9 to CP-8 to get the backsight azimuth, then every layout angle is the difference between the azimuth to the point and that backsight azimuth, measured clockwise.

Inverse is the standard plane computation with the surveying convention: azimuth = atan2(ΔE, ΔN), easting first, north as the first coordinate.

ΔN = 5050.00 − 5150.00 = −100.00 ft; ΔE = 7180.00 − 7600.00 = −420.00 ft distance CP-9 to CP-8 = √(100.00² + 420.00²) = 431.74 ft azimuth CP-9 to CP-8 = atan2(−420.00, −100.00) = 256°36′27″

Layout angles and distances

Inverse from CP-9 to each curve point. The distance is what the instrument will measure; the angle right is the azimuth to the point less the backsight azimuth, brought into the range 0 to 360 degrees.

Distances run from 301 to 389 ft, all comfortable for a single setup, and the angles sweep 88 degrees, so all seven points are visible from one occupation without re-orienting.

Radial layout from CP-9, backsight CP-8 on azimuth 256°36′27″
StationAzimuth CP-9 to pointAngle right from CP-8Distance (ft)
27+00.00319°54′41″63°18′14″354.88
28+00.00335°49′32″79°13′05″318.40
29+00.00354°07′16″97°30′49″301.15
30+00.0013°01′43″116°25′16″306.55
31+00.0030°20′42″133°44′15″333.48
32+00.0044°54′53″148°18′26″377.11
32+23.91 (PT)47°58′50″151°22′23″389.40

Answer

  • PT = 32+23.91, coordinates N 5410.66, E 7889.29.
  • Curve point coordinates: 27+00.00 at N 5421.50 E 7371.47; 28+00.00 at N 5440.48 E 7469.61; 29+00.00 at N 5449.56 E 7569.16; 30+00.00 at N 5448.66 E 7669.11; 31+00.00 at N 5437.79 E 7768.47; 32+00.00 at N 5417.05 E 7866.26.
  • From CP-9 with a backsight on CP-8, turn 63°18′14″ and tape 354.88 ft for 27+00.00, and so on through 151°22′23″ and 389.40 ft for the PT.
  • The radius point of the curve lies at N 4450.40, E 7610.16.

Check

Every staked point must lie exactly one radius from the radius point. The radius point is the PC offset 1000.00 ft on azimuth 74°12′30″ plus 90°, that is 164°12′30″, giving N 4450.40, E 7610.16. Inversing from there to the PT gives 1000.00 ft, and the same holds for every other row.

The chord from the PC to the PT computed from coordinates is √((5410.66 − 5412.66)² + (7889.29 − 7338.02)²) = 551.27 ft, equal to the long chord C = 2R sin(Δ/2) = 551.27 ft.

The azimuth of the forward tangent at the PT must be the back-tangent azimuth plus Δ: 74°12′30″ + 32°00′00″ = 106°12′30″. The chord azimuth in the last table row is 90°12′30″, which is the back-tangent azimuth plus Δ/2, exactly as the chord of a circular arc requires.

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