Knowledge — SurveyCalc

01 Curve Parameters

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Symbol	Name	Description
R ft / m	Radius	Radius of the circular curve. Larger radius = gentler curve.
T ft / m	Sub Tangent	Distance from BC/PC along the tangent to the intersection point (PI).
E ft / m	External	Distance from PI to the midpoint of the curve, measured perpendicular to the chord.
M ft / m	Midordinate	Distance from the midpoint of the chord to the midpoint of the arc.
C ft / m	Chord	Straight-line distance between BC and EC.
Δ degrees	Delta Angle	Total deflection angle of the curve, measured at the PI.
Δ/2 degrees	Delta/2 Angle	Half of the delta angle. Used in many curve calculations.
L ft / m	Arc Length	Length of the curve measured along the arc from BC to EC.

02 Primary Curve Formulas

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Chord

C = 2 × R × sin(Δ/2)

Chord length from radius and half the delta angle.

Sub Tangent (Tangent)

T = R × tan(Δ/2)

Distance along tangent from BC to PI (or PI to EC).

Midordinate

M = R × (1 − cos(Δ/2))

Distance from chord midpoint to arc midpoint.

External

E = R × (1/cos(Δ/2) − 1)

Distance from PI to the curve, perpendicular to the chord.

Arc Length

L = (π × R × Δ) / 180

Arc length = radius × delta angle in radians.

Degree of Curvature (Arc)

Dₐ = 5729.58 / R

Degree of curvature based on arc definition (arc length = 100).

Degree of Curvature (Chord)

Dₒ = 2 × sin⁻¹(50 / R)

Degree of curvature based on chord definition.

03 Supplementary Calculations

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Radius from Midordinate

R = M / (1 − cos(Δ/2))

Calculate radius when midordinate and delta angle are known.

External from Radius

E = R × (1/cos(Δ/2) − 1)

Calculate external distance from radius and delta angle.

PI to Centre Distance

O→PI = √(R² + T²)

Distance from the curve centre to the point of intersection.

External (Alternative)

E = O→PI − R

External from PI-to-centre distance minus radius.

Delta in Radians

Δ = L / R

Delta angle in radians from arc length and radius.

Delta in Degrees

Δ = (L / R) × (180° / π)

Delta angle in degrees from arc length and radius.

04 Chainage & Deflection

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Begin Curve Station

BC = PI − T

Begin curve chainage from PI station and sub tangent.

End Curve Station

EC = BC + L

End curve chainage from begin curve station and arc length.

Deflection Angle

δ/2 = (90 × l) / (π × R)

Deflection angle (degrees) for a given arc length l measured from BC along the curve.

05 Trigonometry Reference

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Basic Trig Ratios

sin(α) = Opposite / Hypotenuse

cos(α) = Adjacent / Hypotenuse

tan(α) = Opposite / Adjacent

Angle Conversion

Degrees to Radians

rad = deg × (π / 180)

Radians to Degrees

deg = rad × (180 / π)

Law of Sines

sin(α)/a = sin(β)/b = sin(γ)/c

Angle form — use to find an unknown angle.

a/sin(α) = b/sin(β) = c/sin(γ)

Distance form — use to find an unknown side.

Law of Cosines

Find side a

a² = b² + c² − 2bc × cos(α)

Find side b

b² = a² + c² − 2ac × cos(β)

Find side c

c² = a² + b² − 2ab × cos(γ)

Find angle α

cos(α) = (b² + c² − a²) / (2bc)

Find angle β

cos(β) = (a² + c² − b²) / (2ac)

Find angle γ

cos(γ) = (a² + b² − c²) / (2ab)

06 Distance Calculations

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Horizontal Distance from Slope & Angle

HD = SD × sin(α)

Horizontal distance from slope distance and vertical angle.

Horizontal Distance (Pythagorean)

HD = √(SD² − VD²)

Horizontal distance from slope distance and vertical distance.

Slope Distance

SD = √(HD² + VD²)

Slope distance from horizontal and vertical distances.

Vertical Distance from Zenith

VD = SD × cos(Z)

Vertical distance from slope distance and zenith angle Z.

Pythagorean Theorem

a² + b² = c²

Relationship between sides of a right triangle.

Hypotenuse

c = √(a² + b²)

Calculate hypotenuse from the two shorter sides.

Side from Hypotenuse

a = √(c² − b²)

Calculate one side when hypotenuse and other side are known.

07 Compound Curves

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Compound Curve Parameters

Symbol	Name	Description
R_l ft / m	Large Radius	Radius of the larger-radius sub-curve.
R_s ft / m	Small Radius	Radius of the smaller-radius sub-curve.
Δ degrees	Total Delta	Total deflection angle = Δ_l + Δ_s.
Δ_l degrees	Delta (Large Curve)	Deflection angle for the large-radius sub-curve.
Δ_s degrees	Delta (Small Curve)	Deflection angle for the small-radius sub-curve.
T_long ft / m	Long Tangent	Distance from main PI to BC₁ — the longer of the two main tangents.
T_short ft / m	Short Tangent	Distance from main PI to EC₂ — the shorter of the two main tangents.
T_l ft / m	Sub-tangent (large)	Tangent length along the R_l curve from BC₁ to PCC.
T_s ft / m	Sub-tangent (small)	Tangent length along the R_s curve from PCC to EC₂.
PCC	Point of Compound Curvature	The point where the two sub-curves meet.
BC₁	Begin Curve	Start of the compound curve, on the large-radius side.
EC₂	End Curve	End of the compound curve, on the small-radius side.

Key Relationship

Total Delta Angle

Δ = Δ_l + Δ_s

The total deflection angle is the sum of the two sub-curve delta angles.

Main Tangents

Long Tangent

T_long = [ R_l − R_s·cos(Δ) + (R_l − R_s)·cos(Δ_s) ] / sin(Δ)

Distance from the main PI to BC₁. Requires R_l, R_s, Δ, and Δ_s.

Short Tangent

T_short = [ R_l − R_s·cos(Δ) − (R_l − R_s)·cos(Δ_l) ] / sin(Δ)

Distance from the main PI to EC₂. Requires R_l, R_s, Δ, and Δ_l.

Solving for Delta Angles — arccos form

Δ_l from T_s (arccos)

Δ_l = cos⁻¹( 1 − [ T_s·sin(Δ) − R_s·(1 − cos(Δ)) ] / (R_l − R_s) )

Solve for the large-curve delta angle given T_s, R_l, R_s, and Δ.

Δ_s from T_l (arccos)

Δ_s = cos⁻¹( 1 − [ R_l·(1 − cos(Δ)) − T_l·sin(Δ) ] / (R_l − R_s) )

Solve for the small-curve delta angle given T_l, R_l, R_s, and Δ.

Solving for Delta Angles — arctan form (alternate)

Δ_l from T_s (arctan)

Δ_l = 2·tan⁻¹( [ T_s·sin(Δ) − R_s·(1−cos(Δ)) ] / [ T_l + T_s·cos(Δ) − R_s·sin(Δ) ] )

Alternate form — algebraically equivalent to the arccos version above.

Δ_s from T_l (arctan)

Δ_s = 2·tan⁻¹( [ T_l·sin(Δ) − R_l·(1−cos(Δ)) ] / [ T_s + T_l·cos(Δ) − R_l·sin(Δ) ] )

Alternate form — algebraically equivalent to the arccos version above. (Symmetric counterpart to the Δ_l arctan form — verify against source if results differ.)

Solving for Radii

Large Radius from R_s, T_s, Δ, Δ_l

R_l = R_s + [ T_s·sin(Δ) − R_s·(1−cos(Δ)) ] / (1−cos(Δ_l))

Solve for the large radius when R_s, T_s, Δ, and Δ_l are known.

Small Radius from R_l, T_l, Δ, Δ_s

R_s = R_l − [ R_l·(1−cos(Δ)) − T_l·sin(Δ) ] / (1−cos(Δ_s))

Solve for the small radius when R_l, T_l, Δ, and Δ_s are known.

08 Traverse Coordinates

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Departures from HD & Azimuth

Northing Departure

ΔN = HD × cos(Az)

Northing departure from horizontal distance and azimuth. Az is a full-circle azimuth in degrees (0–360).

Easting Departure

ΔE = HD × sin(Az)

Easting departure from horizontal distance and azimuth.

Delta from Two Points

Northing Delta

ΔN = N₂ − N₁

Northing change: End point minus Start point.

Easting Delta

ΔE = E₂ − E₁

Easting change: End point minus Start point.

Coordinate Propagation

Northing of Point B

N_B = N_A + ΔN

Compute northing of next point from known point and departure.

Easting of Point B

E_B = E_A + ΔE

Compute easting of next point from known point and departure.

09 Traverse Closure & Adjustment

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Closure Error

Linear Error of Closure

LEC = √(ΣΔN² + ΣΔE²)

Resultant misclosure vector from the sum of northing and easting errors.

Ratio of Precision

RoP = ΣHD / LEC

Traverse precision expressed as a ratio (e.g. 1 : 5000 — higher is better).

Compass Rule Adjustment

Correction Factor — Northing

CF_ΔN = ΣΔN / ΣHD

Proportional correction factor distributed across all northing departures.

Correction Factor — Easting

CF_ΔE = ΣΔE / ΣHD

Proportional correction factor distributed across all easting departures.

Compass Rule Correction — Northing

CR = CF_ΔN × HD_ab

Northing correction for leg ab, proportional to its length.

Compass Rule Correction — Easting

CR = CF_ΔE × HD_ab

Easting correction for leg ab, proportional to its length.

10 Line Geometry

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Slope-Intercept Form

y = mx + b

Line expressed using slope m and y-intercept b.

Point-Slope Form

(y − y₁) = m(x − x₁)

Line through a known point (x₁, y₁) with slope m.

Slope of a Line

m = (y₂ − y₁) / (x₂ − x₁)

Gradient calculated from two known points on the line.

General Form

Ax + By + C = 0

Any line in general (standard) form. Solve for x, y, or any coefficient given the others.

! Important Notes

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›All angle calculations assume degrees unless specified as radians.

›Ensure consistent units throughout — all feet or all metres.

›The curve calculator requires 2–3 known parameters to solve for the rest.

›PI = Point of Intersection | BC / PC = Begin Curve | EC / PT = End Curve.

›Use Settings in the Curve Calculator to adjust decimal precision and angle format (DMS / DD).

›Inline calculators: angle inputs use DMS shorthand — e.g. 45.3000 = 45° 30' 00".