True length: L_true = L_meas × (L_actual / L_nominal)

Temperature correction: C_T = α (T_m − T₀) × L   [α = 11.2 × 10⁻⁶ /°C for steel]

Slope correction: C_slope = −h²/(2L)   [subtract; h = height difference, L = slope length]

Sag correction: C_sag = −w²L³/(24P²)   [w = wt/unit length, P = pull in N]

Pull correction: C_P = (P − P₀) L / (AE)   [A = cross-section area, E = elastic modulus]

Combined: Correct length = Measured length + C_T + C_P − C_slope − C_sag

WCB → QB: 0°–90° → N(WCB)°E  |  90°–180° → S(180−WCB)°E  |  180°–270° → S(WCB−180)°W  |  270°–360° → N(360−WCB)°W

Back bearing: BB = FB + 180° (if FB < 180°)  |  BB = FB − 180° (if FB > 180°)

True bearing: = Magnetic bearing + Declination(E)  |  = Magnetic bearing − Declination(W)

HI method: HI = RL_BM + BS  |  RL = HI − staff reading

Check (HI): ΣBS − ΣFS = Last RL − First RL

Check (R&F): ΣBS − ΣFS = ΣRise − ΣFall = Last RL − First RL

Curvature: C = 0.0785 d² m   (d in km, subtract)

Refraction: r = 0.0112 d² m   (add)

Combined C&R: = 0.0673 d² m   (subtract from observed staff reading)

Reciprocal levelling: True diff = (h₁ + h₂)/2  |  Collimation error = (h₁ − h₂)/2

Horizontal sight: D = Ks + C   (K=100, C≈0 modern) → D = 100s

Inclined sight: D = Ks cos²α  |  V = (Ks/2) sin 2α

RL (elevation angle +α): RL_B = RL_axis + V − h

RL (depression angle −α): RL_B = RL_axis − V − h

RL of axis: = RL of ground station + instrument height (hi)

Stadia intercept: s = top hair − bottom hair reading

Tangential method (both elevation): D = S cos α₁ cos α₂ / sin(α₁ − α₂)

Tangential (one elev, one dep): D = S cos α₁ cos α₂ / sin(α₁ + α₂)

Latitude: L = d cos θ   (+ North, − South)

Departure: D = d sin θ   (+ East, − West)

Closing error: e = √(e_L² + e_D²)   where e_L = ΣL, e_D = ΣD

Accuracy: 1 in (Perimeter / e)

Angular misclosure: Obs. sum − (n−2)×180°   (n = number of sides)

Bowditch corr. to L of side i: = −e_L × l_i / P   (P = perimeter)

Transit corr. to L of side i: = −e_L × |L_i| / Σ|L|

Coordinates: E_n+1 = E_n + D_i  |  N_n+1 = N_n + L_i

Area (coordinate method): A = ½|Σ(E_nN_n+1 − E_n+1N_n)|

Tangent length: T = R tan(Δ/2)

Arc length: L = πRΔ/180   (Δ in degrees)

Long chord: LC = 2R sin(Δ/2)

External distance: E = R(sec(Δ/2) − 1)

Mid-ordinate: M = R(1 − cos(Δ/2))

Offset from long chord at dist x from midpoint: O_x = √(R²−x²) − √(R²−(LC/2)²)

Offset from tangent at dist x: O_x = R − √(R²−x²) ≈ x²/(2R)

Rankine deflection angle per chord: δ = (chord × 1719)/R   (minutes)

Degree of curve (chord definition): D = 1718.87/R   (minutes, R in m)

Length (centrifugal acceleration): L = v³/(CR)   [C = 0.3 m/s³, v in m/s]

Length (IRC empirical): L = 2.7v²/R   [v in km/h, R in m]

Shift: s = L²/(24R)

Total tangent length: T_total = (R + s) tan(Δ/2) + L/2

Spiral angle: φ = L/(2R) radians = L × 180/(2πR) degrees

Length (superelevation runoff): L = e × N × v [N = speed factor, e = superelevation]

Design length = Maximum of all three values.

Rate of change of grade: r = (g₂ − g₁)/L   [g in %, L in m]

Elevation at point x: y = y₀ + g₁x + rx²/2

Highest/lowest point (x from start): x = −g₁/r = g₁L/(g₁−g₂)

Summit curve (L > SSD): L = N×SSD²/(√(2h₁)+√(2h₂))²   [h₁=1.2m, h₂=0.15m]

Sag curve (L > SSD): L = N×SSD²/(2H+2SSD×tanα)   [H=0.75m, α=1°]

N = |g₁−g₂|/100 (algebraic difference of grades)

GPS 3D fix: Minimum 4 satellites required

GPS accuracy: Standard ±15 m | DGPS ±1–3 m | RTK ±1–2 cm

NavIC: 7 satellites, covers India + 1500 km, accuracy <20 m

EDM total error: ε = ±√(a² + (b×D)²)   [a = constant error in mm, b = scale error in ppm, D in km]

Atmospheric correction to EDM: C_atm = (N_std − N_actual) × D × 10⁻⁶

Radiometric resolution: Grey levels = 2^(bit depth)

Spatial resolution: Ground pixel size = (altitude × sensor IFOV)