🛣️ Highway & Railway Engineering

Horizontal Curve Calculator

Professional horizontal (circular) curve design tool for highway and railway engineers. Calculate tangent length, curve length, external distance, mid-ordinate, degree of curve, superelevation and the full setting-out deflection table — following AASHTO, NHA Pakistan, IRC and BS standards.

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Horizontal Curve Calculator

PC · PI · PT · Tangent · Curve Length · Degree of Curve · Superelevation · Setting-Out Table

⚙️ Settings

Unit System

Design Standard

📐 Curve Definition

Define curve by

Deflection / Intersection Angle Δ

Angle between the two tangents (deg · min · sec)

Radius R (m) Radius of the circular curve

Degree of Curve Basis Used only when defining by Degree of Curve

PI Station / Chainage (m) Chainage at Point of Intersection

🚗 Design Speed & Superelevation

Design Speed (km/h) Used for min-radius & superelevation checks

Max Superelevation e_max

Side Friction f

Stopping Sight Distance (m) For sight-line setback (auto-filled by speed)

📋 Setting-Out Chord Interval

Chord / Peg Interval (m) Full-chord spacing for the deflection-angle table

📊 Horizontal Curve Results

✅ Calculated

📐 Need the full runoff & transition design (runoff length, tangent runout, edge-of-pavement profile)? → Superelevation Calculator

Point	Chainage	Chord (m)	Deflection δ	Cumulative Δ/2

📖 Method

How Horizontal Curve Calculation Works — Step by Step

A horizontal curve is a circular arc in plan that smoothly connects two straight road or railway alignments (tangents). It allows a gradual change of direction so that vehicles can negotiate the turn safely at the design speed. The geometry is fixed by just two inputs — the deflection angle between the tangents and the radius of the arc.

Establish the Deflection Angle and PI

The Point of Intersection (PI) is where the two tangent lines meet. The deflection angle Δ (also called the intersection angle I) is the angle through which the alignment turns. It is normally measured in the field with a total station.

Choose Radius R or Degree of Curve D

The designer selects a radius R that satisfies the minimum-radius requirement for the design speed. Alternatively the curve is defined by its Degree of Curve D. Arc definition (metric): D = (arc × 180) / (π × R); for a 30 m arc, D = 1718.873 / R.

Compute the Primary Curve Elements

Tangent length T = R·tan(Δ/2), curve length L = π·R·Δ/180, long chord LC = 2R·sin(Δ/2), external distance E = R(sec(Δ/2) − 1) and mid-ordinate M = R(1 − cos(Δ/2)).

Locate PC and PT Stations

The Point of Curvature PC = PI − T (start of curve) and the Point of Tangency PT = PC + L (end of curve). Note PT is found by adding the curve length to PC — never by adding the tangent length twice.

Check Superelevation and Minimum Radius

For the design speed, e + f = V²/(127R) (metric). The required superelevation must not exceed e_max, and R must be ≥ the minimum radius R_min = V²/[127(e_max+f)].

Generate the Setting-Out Deflection Table

For field layout, the curve is pegged at regular chord intervals from PC. The deflection angle for a chord of length c is δ = 1718.873·c / R minutes. Cumulative deflections at PT must equal exactly Δ/2 — the standard field check.

∑ Core Horizontal Curve Formulas

CIRCULAR (SIMPLE) HORIZONTAL CURVE — Δ = deflection angle, R = radius Tangent Length: T = R · tan(Δ/2) Curve Length: L = π · R · Δ / 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) ) Degree of Curve (arc, a-metre arc): D = (a · 180) / (π · R) Degree of Curve (chord c): D = 2 · asin( c / 2R ) PC Station: PC = PI − T PT Station: PT = PC + L Superelevation: e + f = V² / (127 · R) (V km/h, R m) Minimum Radius: R_min = V² / [127 · (e_max + f)] Sight-line Setback: m = R · ( 1 − cos(28.6479 · S / R) ) Deflection per chord c: δ = 1718.873 · c / R (minutes)

🛣️ Reference

Types of Horizontal Curves — Comparison

Highway and railway alignments use four families of horizontal curve depending on the site geometry and speed-transition requirements.

Type	Description	Where Used	Key Note
Simple	Single circular arc of one radius joining two tangents	Most highway & road curves	Defined by Δ and R only
Compound	Two or more arcs of different radii curving the same way	Interchange ramps, hilly terrain	Radii ratio kept ≤ 1.5:1 (AASHTO)
Reverse	Two arcs curving in opposite directions, common tangent	Constrained alignments, channels	Needs straight between for superelevation reversal
Transition (Spiral)	Spiral easing radius from ∞ to R	High-speed roads, all railways	Length L_s from rate of change of radial accel.

For Pakistan NHA motorways (M-1, M-2, M-5…) AASHTO geometric standards apply with e_max = 0.07. Provincial roads commonly reference IRC. Pakistan Railways follows AREMA track-geometry standards. This calculator solves the simple circular curve; compound/reverse curves are solved arc-by-arc using the same elements.

📊 Standards

Minimum Radius & Superelevation — by Design Speed

Indicative minimum radii computed from R_min = V²/[127(e_max+f)] using e_max = 0.07 (NHA) and AASHTO maximum side-friction factors. Always verify the exact value against the official AASHTO Green Book 2018 / NHA Geometric Design Manual for the project authority.

Design Speed	Side Friction f	R_min (e=0.07)	SSD	Typical Use
50 km/h	0.16	≈ 85 m	65 m	Collector / urban
60 km/h	0.15	≈ 125 m	85 m	Arterial road
70 km/h	0.14	≈ 185 m	105 m	Secondary highway
80 km/h	0.14	≈ 240 m	130 m	National highway
90 km/h	0.13	≈ 320 m	170 m	NHA highway
100 km/h	0.12	≈ 415 m	185 m	Motorway / expressway
110 km/h	0.11	≈ 530 m	220 m	Motorway
120 km/h	0.09	≈ 710 m	250 m	M-roads (M-1, M-2)

💡 For Pakistan's motorways at 120 km/h, the minimum radius for e_max=0.07 is roughly 710 m. Anything sharper requires a speed restriction or higher superelevation. NHA typically designs M-road mainline curves well above this minimum for ride comfort.

📝 Worked Example

Simple Horizontal Curve — NHA Highway

Problem: A National Highway alignment deflects through Δ = 36° at PI chainage 1+000.000 m. A radius of R = 300 m is proposed for a design speed of 100 km/h (e_max = 0.07). Find all curve elements, PC/PT stations, and check the radius.

Given:
 Δ = 36°   R = 300 m   PI = 1+000.000   V = 100 km/h   e_max = 0.07

Step 1 — Tangent Length:
 T = R·tan(Δ/2) = 300 · tan(18°) = 300 · 0.32492 = 97.476 m

Step 2 — Curve Length:
 L = π·R·Δ/180 = π·300·36/180 = 188.496 m

Step 3 — Long Chord, External, Mid-Ordinate:
 LC = 2·300·sin(18°)       = 185.410 m
 E  = 300·(sec18° − 1)     = 300·(1.05146 − 1)  = 15.439 m
 M  = 300·(1 − cos18°)     = 300·(1 − 0.95106)  = 14.683 m

Step 4 — Degree of Curve (30 m arc):
 D = 1718.873 / R = 1718.873 / 300 = 5.730°  (5° 43' 48")

Step 5 — Stations:
 PC = PI − T = 1000 − 97.476 = 0+902.524
 PT = PC + L = 902.524 + 188.496 = 1+091.020

Step 6 — Radius Check (100 km/h, f = 0.12):
 R_min = V²/[127(e_max+f)] = 100²/[127·0.19] = 414.6 m
 R = 300 m  <  414.6 m   → INADEQUATE for 100 km/h
 Either raise R to ≥ 415 m, or reduce design speed to ~85 km/h.

ANSWER: T = 97.476 m | L = 188.496 m | LC = 185.410 m
 E = 15.439 m | M = 14.683 m | D = 5.730°
 PC = 0+902.524 | PT = 1+091.020
 Radius FAILS 100 km/h check — increase to 415 m.

💡 Practice

Expert Design Tips for Horizontal Curves

Radius & Speed

Never design below the minimum radius for the speed — round the chosen radius up to a practical value (e.g. 300, 350, 400 m)
For ride comfort, NHA practice keeps mainline radii well above the absolute minimum
On compound curves keep the radius ratio ≤ 1.5 : 1 between adjoining arcs

Superelevation

Apply the full superelevation transition over the spiral or, for simple curves, over a runoff length before PC
Limit e to e_max (0.07 for NHA) — higher values cause slow-vehicle drift on ice/rain
Provide a minimum 0.5% drainage grade across the superelevated section

Sight Distance

Check lateral clearance: the setback m = R(1 − cos(28.6479·S/R)) must be free of obstructions (cut slopes, barriers, walls)
Sharp curves often fail sight distance even when radius passes — clear the inside of the curve

Setting Out

Always close the deflection table: cumulative δ at PT must equal Δ/2 to the second
Peg at 10 m on open ground, 5 m on sharp curves; use sub-chords at PC and PT to hit round chainages
Re-check PI, PC and PT with the total station before earthwork starts

⚠️ For all highway and railway designs, final horizontal curve calculations must be verified and signed by a licensed Civil / Transport Engineer. This calculator is for preliminary design and checking only.

❓ FAQ

Frequently Asked Questions

The deflection angle Δ is the total angle through which the road alignment turns between the two tangents — it is fixed by the route. The degree of curve D describes the sharpness of the arc: the central angle subtended by a standard arc length (30 m in metric arc definition, 100 ft in imperial). A small D means a flat, large-radius curve. They are related by D = 1718.873/R for a 30 m arc.

First compute the tangent length T = R·tan(Δ/2). The Point of Curvature PC = PI − T. Then compute the curve length L = πRΔ/180. The Point of Tangency PT = PC + L. A common mistake is computing PT = PI + T — that gives the wrong chainage because the curve is shorter than two tangent lengths.

Pakistan's National Highway Authority (NHA) follows AASHTO geometric design standards for motorways and national highways, with maximum superelevation e_max = 0.07. Provincial road departments also reference IRC (Indian Roads Congress). Pakistan Railways uses AREMA for track geometry. For Gulf projects, AASHTO or BS standards apply depending on the client authority.

From the point-mass equation e + f = V²/(127R) in metric units (V in km/h, R in metres). Rearranged for the sharpest allowable curve: R_min = V²/[127(e_max + f_max)], where f_max is the maximum side-friction factor from AASHTO (it decreases as speed rises). For 100 km/h with e=0.07 and f=0.12, R_min ≈ 415 m.

Superelevation (e) is the banking of the carriageway toward the inside of the curve so that part of the centrifugal force is balanced by gravity. AASHTO/NHA limit e to 0.07 on highways (snow-free, high traffic), up to 0.08–0.10 on rural roads, and IRC allows up to 0.10 in hilly areas. The remainder of the force is resisted by side friction f.

In the deflection-angle (Rankine) method of setting out, the angle from the back tangent to the chord to any point equals half the angle subtended by the arc to that point. At the end of the curve (PT) the whole arc subtends Δ at the centre, so the deflection from the tangent is exactly Δ/2. If your cumulative deflection at PT does not equal Δ/2, there is an arithmetic error in the chord lengths.

Horizontal Curve Calculator