What's degree of curve D?

US arc definition: D is the angle subtended by 100 ft of arc. So D = 5729.578/R (with R in ft). Railroad uses chord definition (angle subtended by 100-ft chord). They converge for large R but differ for sharp curves.

What's the difference between PC, PT, and PI?

PC = Point of Curvature (start). PT = Point of Tangent (end). PI = Point of Intersection (where the two tangents would meet if curve weren't there). Δ is the deflection angle measured at PI.

What's required lateral clearance for sight distance?

M_ssd = R[1 − cos(28.65 SSD/R)] (US: ft, deg). Compare to actual mid-ordinate distance from inside of curve to obstruction. If actual M < M_ssd, increase R, remove obstruction, or accept reduced design speed.

Horizontal Curve Geometry

Circular horizontal curve elements: radius, degree of curvature (arc and chord definitions), tangent T, arc length L, mid-ordinate M, external E. Plus AASHTO horizontal sight-distance lateral clearance check.

Units

Input mode

R (or D)ft (R) — or degrees if D mode

Deflection angle, Δdegrees (positive = curve to right)

SSD (for clearance check)ft (= AASHTO SSD at design speed)

Radius, R—ft

Degree of curve, D (arc)—degrees

Tangent length, T—ft (PT to PI)

Arc length, L_c—ft (PT to PC along curve)

Long chord, LC—ft

Mid-ordinate, M—ft

External distance, E—ft

Required lateral clearance, M_ssd—ft (from inside edge of travelway)

Defaults: R = 1000 ft, Δ = 30°. AASHTO horizontal sight-distance clearance: M = R [1 − cos(28.65 SSD/R)]; check against actual roadside obstruction.

$$ T = R \tan(\Delta / 2), \quad L_c = R \cdot \Delta_{rad}, \quad LC = 2R \sin(\Delta/2) $$

$$ M = R \left(1 - \cos\frac{\Delta}{2}\right), \quad E = R \left(\sec\frac{\Delta}{2} - 1\right) $$

Degree of curve, arc definition (US): D × R = 5729.578 ft·deg (at 100 ft of arc).

$$ D_{arc} = \frac{5729.578}{R \text{ (ft)}} $$

AASHTO horizontal SSD lateral clearance:

$$ M_{ssd} = R \left[1 - \cos\!\left(\frac{28.65 \, \text{SSD}}{R}\right)\right] $$

R radius · D degree of curve (US arc def: angle subtended by 100 ft of arc) · T tangent (PI to PC or PI to PT) · L_c arc length · LC long chord · M mid-ordinate (PC-PT chord to curve) · E external distance (PI to curve) · Δ deflection (intersection) angle.

Curve labels — PC, PT, PI

PC = Point of Curvature (start of curve). PT = Point of Tangent (end of curve). PI = Point of Intersection (where the two tangent lines would meet if the curve weren't there). Δ = deflection angle = the angle the alignment turns through, measured at PI.

Arc vs chord definition of D

US highway design uses arc definition: D is the angle subtended by 100 ft of curve arc. Railroad design historically used chord definition: D is the angle subtended by a 100-ft chord. They differ for sharp curves but converge for large radii (R > 500 ft). This calculator uses arc definition.

Sight distance on horizontal curves

Drivers can't see around obstructions on the inside of a horizontal curve. Required lateral clearance from the centerline of the inside lane to any obstruction:

M_ssd = R [1 − cos(28.65 × SSD / R)] (US: ft, degrees)

Compare against the horizontal offset to actual obstruction (median barrier, building, hillside, sound wall). If actual M < M_ssd, you'll need to either remove the obstruction, increase R, or accept reduced design speed.

Note: M_ssd is measured to the driver eye position (3.5 ft from inside edge of travelway, on average), not to the inside edge.

Spirals and superelevation transitions

This calculator is for circular curves only. Most highway design includes spirals (clothoid curves) at PC and PT to gradually introduce/exit centripetal acceleration and superelevation. The spiral's L_s length is approximately L_s = 1.6 V³ / R for design speed V (mph) and R (ft). Add this to the geometric design — most CAD survey software handles it automatically.

Reverse and compound curves

Compound curves (two consecutive curves in the same direction with different radii) and reverse curves (consecutive curves in opposite directions) are common in mountain alignments. AASHTO recommends a tangent of at least 200 ft (60 m) between reverse curves to allow superelevation transition.

Reference: AASHTO (2018). A Policy on Geometric Design of Highways and Streets, 7th ed., §3.3. Garber, N.J., Hoel, L.A. (2015). Traffic and Highway Engineering, 5th ed., Cengage, ch. 16. Hickerson, T.F. (1967). Route Surveying and Design, 5th ed., McGraw-Hill.