Vertical Curve Calculator
Design and analyze vertical curves for roadway and railway projects. Enter the incoming grade (G₁), outgoing grade (G₂), curve length (L), PVI station, and PVI elevation. Choose the station interval for detailed output. Results include BVC and EVC station/elevation, K value, rate of grade change, algebraic grade difference, maximum and minimum elevations, critical high/low points, and a complete station-by-station elevation table. The parabolic formula accounts for both crest curves (hill tops) and sag curves (valley bottoms) per AASHTO Green Book standards.
Approach grade (positive = uphill, negative = downhill)
Departure grade (positive = uphill, negative = downhill)
Total horizontal length of the vertical curve
Station of the Point of Vertical Intersection (in feet)
Elevation at the PVI in feet
Y = Y_bvc + G₁X + (G₂−G₁)X² / (2L)
Offset from tangent:
e = X² × A / (200L)
Where:
Y = Elevation at distance X from BVC
Y_bvc = Elevation at BVC
G₁ = Grade in (%)
G₂ = Grade out (%)
L = Curve length (ft)
X = Distance from BVC (ft)
A = |G₁ − G₂| — algebraic grade difference
K = L / A (design speed parameter)
BVC Station = PVI Station − L/2
EVC Station = PVI Station + L/2
BVC Elevation = PVI Elev − G₁ × L/2
G₁ = +2.5%, G₂ = −1.5%, L = 400 ft
PVI at Sta 1,200 ft, Elev = 500.00 ft
BVC: Sta 1,000 ft, Elev = 495.00 ft
EVC: Sta 1,400 ft, Elev = 497.00 ft
A = |2.5 − (−1.5)| = 4.0%
K = 400 / 4.0 = 100 ft/%
At Sta 1,100 ft (X = 100 ft from BVC):
Y = 495.00 + 0.025(100) + (−0.015−0.025)×100²/(2×400)
Y = 495.00 + 2.50 − 0.50 = 497.00 ft
High point near Sta 1,160 ft, Elev ~497.19 ft
What is a vertical curve in road design?
A vertical curve is a parabolic transition between two roadway grades, used to provide smooth and safe travel between different slopes. Sag curves connect a downhill grade to an uphill grade (or less downhill). Crest curves connect an uphill to a downhill grade. Key points: BVC (Beginning of Vertical Curve), PVI (Point of Vertical Intersection), and EVC (End of Vertical Curve). The curve is designed using parabolic equations to ensure constant rate of change of grade, providing comfortable riding conditions and adequate sight distance for drivers.
How is the elevation at any point on a vertical curve calculated?
Using the parabolic formula: Y = Y_bvc + G₁×X + (G₂−G₁)×X²/(2L), where Y_bvc is the elevation at BVC, G₁ is grade in (decimal), X is distance from BVC, G₂ is grade out (decimal), and L is curve length. The offset from the tangent (e) = X² × A / (200L), where A = |G₁ − G₂|. Maximum offset occurs at the midpoint or at the PVI depending on grade difference. For symmetric curves, the highest/lowest point occurs at distance X = G₁×L / (G₁−G₂) from BVC when G₁ and G₂ have opposite signs.
What is the difference between a crest and a sag vertical curve?
A crest vertical curve occurs when G₁ is positive and G₂ is negative (top of a hill). On a crest curve, sight distance is critical — drivers cannot see over the crest. Minimum curve length is determined by stopping sight distance (SSD) or passing sight distance. A sag vertical curve occurs when G₁ is negative and G₂ is positive (bottom of a valley). On sag curves, sight distance is limited by headlight illumination at night and overhead structures (bridges). Sag curves are also critical for drainage — the low point should not be at a catch basin location where ponding may occur.
What is the minimum length for a vertical curve?
Minimum vertical curve length depends on: 1) Design speed — higher speeds require longer curves. For a 50 mph road: minimum K value (length per % grade change) = 56 for crest curves, 46 for sag curves. K = L / A where A = |G₁ − G₂|. For A = 4% at 50 mph: L_min = K × A = 56 × 4 = 224 ft for crest, 46 × 4 = 184 ft for sag. 2) Comfort — maximum vertical acceleration of 1 ft/s². 3) Drainage — minimum 100 ft length for drainage (max grade change of 0.5% within 50 ft of low point). 4) Aesthetics — longer curves generally look better.
🔗 Related Calculators
📐 Formula
Y = Y_bvc + G₁X + (G₂−G₁)X² / (2L)
Offset from tangent:
e = X² × A / (200L)
Where:
Y = Elevation at distance X from BVC
Y_bvc = Elevation at BVC
G₁ = Grade in (%)
G₂ = Grade out (%)
L = Curve length (ft)
X = Distance from BVC (ft)
A = |G₁ − G₂| — algebraic grade difference
K = L / A (design speed parameter)
BVC Station = PVI Station − L/2
EVC Station = PVI Station + L/2
BVC Elevation = PVI Elev − G₁ × L/2
📝 Example Calculation
G₁ = +2.5%, G₂ = −1.5%, L = 400 ft
PVI at Sta 1,200 ft, Elev = 500.00 ft
BVC: Sta 1,000 ft, Elev = 495.00 ft
EVC: Sta 1,400 ft, Elev = 497.00 ft
A = |2.5 − (−1.5)| = 4.0%
K = 400 / 4.0 = 100 ft/%
At Sta 1,100 ft (X = 100 ft from BVC):
Y = 495.00 + 0.025(100) + (−0.015−0.025)×100²/(2×400)
Y = 495.00 + 2.50 − 0.50 = 497.00 ft
High point near Sta 1,160 ft, Elev ~497.19 ft