Vertical (Parabolic) Curves - Cheat Sheet
Your last-minute revision companion for vertical curve design, elevation calculations, turning points, and sight-distance requirements. Master the parabolic formula, grade sign conventions, and exam pitfalls in 30 minutes.
Fundamental Parabolic Curve Equation
Formulas
y = elev_PC + g₁·x + (r/2)·x²
Meaning
y = elevation at distance x from PC; elev_PC = elevation at point of curvature; g₁ = initial grade (decimal); x = horizontal distance from PC (m); r = rate of grade change per unit length
Watch out
Grade must be in decimal form (3% = 0.03, −2% = −0.02). Distance x must be measured FROM THE PC, not from PI or PT. Maximum x is L (curve length).
When to use
Always — this is the master elevation equation. Use it to find any elevation along the curve.
r = (g₂ − g₁) / L
Meaning
r = constant rate of grade change per meter of length; g₂ = final (exit) grade (decimal); g₁ = initial (entry) grade (decimal); L = total length of vertical curve (m)
Watch out
Sign of r matters: negative r = crest (summit), positive r = sag (valley). Many students flip signs. Write r with its sign explicitly.
When to use
First step in any vertical curve problem. Determines if curve is crest (r < 0) or sag (r > 0).
Important facts
- The vertical curve is PARABOLIC: second derivative (d²y/dx²) is constant = r, not linear.
- For a symmetrical curve, the PI is at station PC + L/2; the maximum vertical offset occurs here.
- Grades must always be written with sign: +3%, −2%, +0.5%, etc. Never drop the sign.
- The curve connects two tangent grades; the grades themselves don't change slope—only the transition does.
- In the elevation formula y = elev_PC + g₁·x + (r/2)·x², the first term is the PC elevation, the second is the linear ramp, and the third is the parabolic correction.
Key definitions
Vertical Parabolic Curve
Example
A +3% grade smoothly transitions to −2% grade over 200 m via a parabolic curve.
A smooth, parabolic-shaped connection between two tangent grades (g₁ and g₂) with constant rate of grade change, used in road profile design for comfort, sight distance, and drainage.
Rate of Grade Change (r)
Example
r = (−2% − 3%) / 200 m = −0.0005/m = −0.05%/m
The change in grade per unit length of curve; constant for a parabola, measured in m⁻¹ or %/m.
Crest Curve
Example
+2% → −3%; sight line blocked by hump; limited by stopping distance.
A vertical curve where g₁ > g₂ (upgrade to downgrade); forms a summit; common sight-distance concern.
Sag Curve
Example
−4% → +1.5%; night visibility and water drainage are design concerns.
A vertical curve where g₁ < g₂ (downgrade to upgrade); forms a valley; limited by headlight throw and drainage.
PC (Point of Curvature) / BVC (Beginning of Vertical Curve)
Example
Station 10+000 m, elevation 100.00 m, grade +3%.
Start of the vertical curve; where the initial tangent grade g₁ meets the parabola.
PT (Point of Tangency) / EVC (End of Vertical Curve)
Example
Station 10+200 m (after 200 m curve length), elevation given by formula.
End of the vertical curve; where the parabola meets the final tangent grade g₂.
PI (Point of Intersection)
Example
For PC at 10+000 and L = 200 m, PI is at 10+100 m.
Theoretical intersection of the two tangent grades (before curving); for symmetrical curves, PI is at midpoint station.
Diagrams to know
- Crest curve profile: tangent from lower left at +3%, curves down to a summit, then continues down at −2%. Mark PC, PT, PI, summit location (x from PC).
- Sag curve profile: tangent from upper left at −4%, curves down to low point, then curves back up at +1.5%. Mark low point location.
- Vertical offset diagram: parabola above or below tangent line; offset = (r/2)·x² in magnitude.
Turning Point (Summit or Low Point) Location
Formulas
x = (g₁·L) / (g₁ − g₂)
Meaning
x = horizontal distance from PC to the turning point (summit for crest, low point for sag) (m); L = curve length (m)
Watch out
The denominator is (g₁ − g₂), NOT (g₂ − g₁). If you reverse it, you get negative distance (wrong direction). Check that 0 < x < L; if not, the turning point is outside the curve.
When to use
When asked 'where is the summit?' or 'where is the low point?'. Also written as x = −g₁/r.
Important facts
- For a crest (g₁ > g₂), the turning point is the SUMMIT: a maximum.
- For a sag (g₁ < g₂), the turning point is the LOW POINT: a minimum.
- If x < 0 or x > L, the turning point lies outside the curve; the curve is monotonic (only ascending or only descending).
- At the turning point, the gradient dy/dx = g₁ + r·x = 0, which yields x = −g₁/r.
- Many textbooks use x = (|g₁|·L) / (|g₁| + |g₂|) or x = (g₁·L) / (g₁ − g₂); verify your formula matches the sign convention.
Key definitions
Turning Point Elevation (Summit or Low Point)
Example
At x = 120 m, if y_PC = 100.00 m, g₁ = 0.03, r = −0.00025/m, then y_summit = 100 + 0.03(120) + (−0.00025/2)(120)² = 101.8 m.
The elevation where the slope is zero (dy/dx = 0); found by substituting the turning-point station into the elevation formula.
Diagrams to know
- Grade line diagram: draw g₁ from PC, g₂ from PT; they cross at PI. The parabola sits between the two tangent lines.
- Turning-point location: mark x on the horizontal axis from PC; show it between 0 and L.
Reactions or equations
dy/dx = g₁ + r·x = 0 ⟹ x = −g₁/r = (g₁·L) / (g₁ − g₂)
Note
This is the calculus foundation: differentiate y = elev_PC + g₁·x + (r/2)·x² to get dy/dx = g₁ + r·x. Set equal to zero for turning point.
Conditions
At the turning point, the grade (slope) is zero.
Elevation Calculations Along the Curve
Formulas
Δy = g₁·x + (r/2)·x²
Meaning
Δy = vertical rise from PC elevation at distance x (m); same as (y − elev_PC)
Watch out
This is rise from PC only. Total elevation = elev_PC + Δy. Sign of Δy can be positive (rise) or negative (drop).
When to use
When PC elevation is 0 or when you only need the elevation change (not absolute elevation).
y_PT = elev_PC + g₁·L + (r/2)·L²
Meaning
y_PT = elevation at end of curve (PT); plug x = L into the master equation.
Watch out
y_PT is NOT simply elev_PC + (g₁ + g₂)/2 · L. The parabolic correction (r/2)·L² is crucial.
When to use
To check: y_PT should also equal (elev_PC + g₂·L) if the second tangent extends backward from PT, confirming the formula.
y = elev_PC + (g₁ + g₂)/2 · x + (r/2)·x·(x − L)
Meaning
Alternative form emphasizing symmetry: uses average grade plus parabolic term relative to curve length.
Watch out
Do NOT mix forms in the same problem. Stick to y = elev_PC + g₁·x + (r/2)·x².
When to use
Rarely; shown here for completeness. Standard form (y = elev_PC + g₁·x + (r/2)·x²) is preferred.
Important facts
- At x = 0 (the PC), y = elev_PC (the curve touches the initial tangent).
- At x = L (the PT), y = elev_PC + g₁·L + (r/2)·L². This must equal the elevation reached by following g₂ from the PI backward.
- For intermediate stations (e.g., 25 m intervals), use the master formula directly.
- The parabolic term (r/2)·x² is usually small in magnitude; omitting it is a common exam mistake that gives wrong answers to 1–2 decimal places.
Key definitions
Vertical Offset (from Tangent)
Example
At x = 100 m on a curve with r = −0.0005/m, offset = (−0.0005/2)·(100)² = −0.0025 m = −2.5 mm below tangent.
Perpendicular distance between the parabolic curve and the tangent line (initial grade g₁); equals (r/2)·x² in magnitude.
Diagrams to know
- Elevation profile showing PC, summit/low point, and PT with annotated elevations.
- Table of stations vs. elevations (e.g., every 20 m or 50 m interval along the curve).
Vertical Offset and Mid-Curve Properties
Formulas
offset(x) = (r/2)·x²
Meaning
Vertical distance from the initial tangent line (grade g₁) to the parabola at distance x (m); positive is above tangent, negative is below.
Watch out
Sign: For a crest (r < 0), the offset is negative (curve below tangent). For a sag (r > 0), the offset is positive (curve above tangent). Do not confuse direction.
When to use
To find how much the curve deviates from the straight-line grade at any point.
offset_max = |r|·L² / 8 = |A|·L / 800
Meaning
Maximum vertical offset, occurring at the midpoint (x = L/2) of a symmetrical curve; A = |g₂ − g₁| (%) is the absolute algebraic grade difference.
Watch out
This formula uses absolute values (|r|, |A|). The sign indicates direction (below tangent for crest, above for sag), but the magnitude is always positive here.
When to use
Quick check: how much sag or crest does this curve have? Also appears in sight-distance formulas.
Important facts
- At x = L/2, the offset is (r/2)·(L/2)² = r·L² / 8, which equals the maximum offset in magnitude.
- For a symmetrical curve, the turning point is NOT at x = L/2 unless g₁ = −g₂. For an unsymmetrical curve (e.g., g₁ = 3%, g₂ = −2%), the turning point is off-center.
- The offset formula (r/2)·x² shows that the curve is parabolic: doubling x quadruples the offset.
- At the PT (x = L), the curve touches the second tangent line (grade g₂); the offset relative to the first tangent is (r/2)·L², but relative to g₂ it is zero.
Key definitions
Maximum Vertical Offset
Example
For L = 200 m and A = 5%, offset_max = 5 × 200 / 800 = 1.25 m.
The perpendicular distance between the parabolic curve and the initial tangent grade, measured at the midpoint of the curve (x = L/2).
Diagrams to know
- Offset curve diagram: parabola sits above or below a horizontal tangent, with maximum offset marked at x = L/2.
Sight Distance and Vertical Curve Length
Formulas
L = (A·S²) / (200·(√h₁ + √h₂)²) [when S < L]
Meaning
L = minimum crest-curve length for stopping sight distance (m); A = |g₁ − g₂| (algebraic grade difference in %); S = stopping sight distance (m); h₁ = eye height (m, typically 1.08 m); h₂ = object height (m, typically 0.6 m for passenger car).
Watch out
This formula applies when S < L. If S ≥ L (very long curve), use the alternative formula L = 2·S − (200·(√h₁ + √h₂)²) / A. Also, A must be in percent (not decimal): a 5% grade change is 5, not 0.05.
When to use
When designing a CREST curve governed by line-of-sight over the hump. S < L means the stopping distance is shorter than the curve.
L = (A·S²) / (400·h) [Sag curve, headlight]
Meaning
L = minimum sag-curve length for headlight sight distance (m); h = headlight height (m, typically 0.6 m); other parameters as above.
Watch out
Different formula than crest; sag curves use a single object height (headlight), not two heights. Make sure you're not using the crest formula for a sag problem.
When to use
When designing a SAG curve at night, limited by headlight beam throw and angle (typically 1° above horizontal).
A = |g₂ − g₁| × 100
Meaning
A = algebraic grade difference in percent (%); must convert decimal grades to percentages (e.g., g₁ = 0.03 ⟹ 3%).
Watch out
Easy mistake: forgetting to multiply by 100 or using decimals instead of percentages. If A comes out as 0.05, you've used decimals; multiply by 100 to get A = 5%.
When to use
Every sight-distance calculation. Always convert grade decimals to percentages first.
Common values
Value
1.08 m
Symbol
h₁
Quantity
Standard Driver's Eye Height
Value
0.6 m
Symbol
h₂
Quantity
Standard Object Height (Passenger Car)
Value
0.6 m (eff.)
Symbol
h
Quantity
Standard Headlight Height
Value
~1° above horizontal
Symbol
θ
Quantity
Headlight Beam Angle
Important facts
- NSCP 2015 specifies minimum design speeds and corresponding stopping sight distances (e.g., 60 km/h → 120 m SSD).
- Crest curves are more restrictive in rural areas where speed is high; sag curves are more critical at night or on grades with poor drainage.
- The formulas use √h₁ + √h₂ (not h₁ + h₂), which changes results significantly.
- For standard passenger-car design: h₁ = 1.08 m (driver's eye), h₂ = 0.6 m (object on roadway), headlight h ≈ 0.6 m (effective height of illumination).
- If your calculated L is negative or very small, the curve sight distance is NOT the controlling factor; drainage, comfort, or road geometry may control L instead.
Key definitions
Stopping Sight Distance (SSD)
Example
At 60 km/h, typical SSD ≈ 120 m (per NSCP and local standards).
The minimum distance a vehicle can travel and stop safely upon sighting an obstruction, including reaction time and braking distance.
Crest Curve Sight-Distance Requirement
Example
A 5% crest curve with SSD = 120 m requires L ≥ 54 m (approx.).
The vertical curve length must be long enough so that a driver at eye height h₁ can see an object of height h₂ over the hump, within the stopping sight distance.
Sag Curve Sight-Distance Requirement (Headlight)
Example
A 5% sag curve with SSD = 120 m requires L ≥ 90 m (approx.) for night visibility.
The vertical curve length must be long enough so that headlights can illuminate the roadway ahead within the stopping sight distance.
Diagrams to know
- Crest curve sight-distance diagram: draw eye at h₁, object at h₂, line of sight tangent to the parabola; mark S and summit.
- Sag curve headlight diagram: draw headlight beam at h above road, angled at ~1° above horizontal, show minimum curve length for beam to reach S ahead.
Grade Sign Conventions and Curve Classification
Important facts
- ALWAYS use signed grades: +, −, or 0. Dropping the sign is the #1 source of errors.
- r = (g₂ − g₁) / L: a NEGATIVE r indicates a crest; a POSITIVE r indicates a sag.
- For a +2% to −3% curve: g₁ = +0.02, g₂ = −0.03, so r = (−0.03 − 0.02) / L = −0.05 / L < 0 (crest).
- For a −4% to +1.5% curve: g₁ = −0.04, g₂ = +0.015, so r = (0.015 − (−0.04)) / L = 0.055 / L > 0 (sag).
- In a crest curve, the parabola curves downward (concave down); in a sag curve, it curves upward (concave up).
Key definitions
Positive Grade (+)
Example
+4% grade on an approach to a bridge.
Upward slope in the direction of increasing station; +3% means 3 m rise per 100 m horizontal.
Negative Grade (−)
Example
−3% grade leaving a hill.
Downward slope in the direction of increasing station; −2% means 2 m drop per 100 m horizontal.
Crest Vertical Curve
Example
Hilltop intersection; line-of-sight limited.
Formed when initial grade g₁ > final grade g₂ (e.g., +3% to −2%); creates a summit; r < 0.
Sag Vertical Curve
Example
Valley crossing; drainage and headlight visibility matter.
Formed when initial grade g₁ < final grade g₂ (e.g., −4% to +1%); creates a valley; r > 0.
Diagrams to know
- Crest vs. sag profile overlay: two curves side by side, one with summit above tangent lines, one with valley below.
- Sign diagram: vertical axis labeled with +g and −g; arrows showing direction of positive and negative grades.
Problem-Solving Flowchart and Station–Elevation Tables
Important facts
- Always tabulate elevations at regular intervals (matching field measurements or construction templates).
- Station numbers are usually written as (hundreds) + (ones), e.g., Sta. 12+345 = 12,345 m.
- When solving a problem, first calculate r, then the turning point location (x), then key elevations (PC, turning point, PT), then fill in intermediate stations.
- Round elevations to 0.01 m (cm) unless the problem specifies otherwise.
- Check your work: the elevation at PT calculated via the curve should match the elevation reached by the initial grade plus the curve rise/fall.
Key definitions
Station
Example
PC at Sta. 5+200, PT at Sta. 5+400 (curve length L = 200 m).
Horizontal distance along the roadway centerline, typically measured from a fixed origin (e.g., 0+000, 10+500, where 10+500 = 10,500 m).
Station–Elevation Table
Example
Sta. | Elev. (m)
5+200 | 100.00
5+220 | 100.54
5+240 | 101.04
A tabulated list of elevations at regular station intervals (e.g., every 20 m, 50 m) for construction staking and profile drawing.
Diagrams to know
- Station–elevation table template: columns for Station, Distance from PC, Grade (%), and Elevation (m).
- Profile sketch: plot stations on x-axis, elevations on y-axis; draw tangent lines and parabolic curve.
Must remember
- GRADE SIGNS ARE CRITICAL: Always write +3%, −2%, etc. Never drop the sign. Decimals must also be signed: +0.03, −0.02. This is the #1 source of errors.
- The Master Elevation Formula: y = elev_PC + g₁·x + (r/2)·x². This single formula solves 80% of exam problems. Memorize it exactly, and verify that x is measured FROM PC, not from PI or PT.
- Rate of Grade Change: r = (g₂ − g₁) / L. Negative r = crest (summit); positive r = sag (valley). The sign of r tells you the curve type and the direction of the parabola.
- Turning Point Location: x = (g₁·L) / (g₁ − g₂) = −g₁/r. This is where dy/dx = 0. Always check that 0 < x < L; if outside this range, the turning point is off the curve.
- Sight-Distance Formulas Are Different: Crest uses √h₁ + √h₂ (two heights); sag uses single h (headlight). Do NOT mix them. If the problem says 'headlight' or 'night visibility,' it's a sag curve.
- Algebraic Grade Difference A = |g₂ − g₁| × 100 (in percent, not decimal). Common mistake: A = 0.05 instead of A = 5%. Always multiply by 100 in sight-distance calcs.
- Symmetry is NOT Always Achieved: For g₁ ≠ −g₂, the turning point is NOT at x = L/2. The curve is parabolic, not symmetric about the vertical axis (unless |g₁| = |g₂|).
- Vertical Offset Direction: For a crest (r < 0), offset is below the initial tangent (negative); for a sag (r > 0), offset is above (positive). The magnitude is always (r/2)·x² or |r|·L²/8.
- The PT Elevation Check: Calculate y_PT via the curve formula: y_PT = elev_PC + g₁·L + (r/2)·L². This MUST equal the elevation if you follow grade g₂ backward from PI. If it doesn't, you have an arithmetic error.
- Exam Trick: If the problem gives you PC station/elevation and a curve length, and asks for PT elevation—do NOT simply add L × (average grade). Use the full parabolic formula, or you will be off by the parabolic correction term.
Last minute tips
- Always draw a sketch (even rough) of the vertical curve: show PC, PI, PT, g₁, g₂, and mark whether it's a crest or sag. This 30-second sketch prevents sign errors and station mix-ups.
- If a problem says 'crest' or mentions 'summit,' immediately think r < 0 and use the eye–object sight formula L = (A·S²) / (200·(√h₁ + √h₂)²). If it says 'sag' or 'headlight,' think r > 0 and use L = (A·S²) / (400·h).
- Before plugging numbers into the elevation formula y = elev_PC + g₁·x + (r/2)·x², STOP and verify: (i) g₁ and g₂ have signs, (ii) x is measured from PC (not PI), (iii) r has the correct sign (r = (g₂ − g₁)/L, order matters).
- When calculating the turning point x = (g₁·L)/(g₁ − g₂), use DECIMALS for grades (0.03, not 3%), not percentages. If you use 3 and −2 directly, you'll get wrong units. Also, check that x > 0 and x < L; if not, the curve has no turning point.
- For sight-distance problems, write down A = |g₂ − g₁| × 100 as a separate step BEFORE substituting into L formulas. This prevents the common mistake of forgetting the 100 multiplier or using decimals instead of percentages.
Comparison tables
Crest vs. Sag Vertical Curves
| Property | Crest Curve | Sag Curve |
|---|---|---|
| Grade Relationship | g₁ > g₂ | g₁ < g₂ |
| Rate of Grade Change (r) | r < 0 (negative) | r > 0 (positive) |
| Curve Shape | Concave down; summit (high point) | Concave up; valley (low point) |
| Parabola vs. Tangent | Curve lies below initial tangent (offset negative) | Curve lies above initial tangent (offset positive) |
| Primary Sight-Distance Issue | Line of sight blocked by hump; limited by eye and object heights; day visibility | Headlight throw limited; night visibility; also consider drainage |
| Typical Design Control | Stopping sight distance (SSD); L = A·S² / (200·(√h₁ + √h₂)²) | Headlight distance (night); L = A·S² / (400·h); also comfort and drainage |
| Turning Point (if within curve) | Summit (maximum elevation); dy/dx = 0 | Low point (minimum elevation); dy/dx = 0 |
| Example | +3% to −2% over 200 m | −4% to +1.5% over 160 m |
Vertical Curve Formula Selection Guide
| Question Type | Formula(s) to Use | Key Input | Common Pitfall |
|---|---|---|---|
| Find elevation at distance x from PC | y = elev_PC + g₁·x + (r/2)·x² | elev_PC, g₁ (decimal), x, r (sign matters) | Forgetting the (r/2)·x² term or using x from PI instead of PC |
| Find location of summit or low point | x = (g₁·L) / (g₁ − g₂) | g₁, g₂ (decimals), L | Reversing denominator to (g₂ − g₁); forgetting to check 0 < x < L |
| Find elevation at PT | y_PT = elev_PC + g₁·L + (r/2)·L² | elev_PC, g₁, L, r | Using simple average grade (g₁ + g₂)/2 instead of accounting for parabola |
| Find maximum vertical offset | offset_max = |r|·L² / 8 or |A|·L / 800 | r or A (%), L | Forgetting the absolute value; confusing magnitude with direction (above vs. below tangent) |
| Find minimum curve length (crest, SSD) | L = (A·S²) / (200·(√h₁ + √h₂)²) [if S < L] | A (%), S (m), h₁ (m), h₂ (m) | Using A as decimal instead of percent; wrong formula (sag headlight) for crest; not checking S < L assumption |
| Find minimum curve length (sag, headlight) | L = (A·S²) / (400·h) [if S < L] | A (%), S (m), h (m) | Using crest formula; forgetting h is headlight height (0.6 m), not eye height |
| Find rate of grade change | r = (g₂ − g₁) / L | g₁, g₂ (decimals), L | Forgetting signs of g₁ and g₂; order matters (g₂ first in numerator) |
Common Exam Values and Standards
| Parameter | Standard Value | Units | Notes |
|---|---|---|---|
| Driver's Eye Height | 1.08 | m | NSCP 2015; used in crest SSD formula |
| Object Height (Passenger Car) | 0.6 | m | NSCP 2015; used in crest SSD formula |
| Headlight Height (Effective) | 0.6 | m | NSCP 2015; used in sag headlight formula |
| Headlight Beam Angle | ~1° | above horizontal | Typical; affects sag curve design |
| Min. Stopping SSD @ 60 km/h | 120 | m | NSCP 2015 / PRC exam typical |
| Min. Stopping SSD @ 80 km/h | 160 | m | NSCP 2015 / PRC exam typical |
| Min. Stopping SSD @ 100 km/h | 200 | m | NSCP 2015 / PRC exam typical |
Diagrams
Vertical Curve Problem-Solving Decision Tree
Decision tree showing which formula to use based on the question type (elevation calculation, turning point location, or sight-distance curve length). Use this to quickly identify the correct approach in an exam.
Vertical Parabolic Curves: Formula Map and Relationships
Mind map organizing all key formulas and their relationships. Use this to quickly find the right formula and understand dependencies (e.g., r depends on g₁, g₂, L; elevation depends on r and x).
Crest vs. Sag Curve Identification and Design
Step-by-step flowchart to identify whether a curve is a crest or sag, understand its geometry, and select the correct sight-distance formula. Essential for exam accuracy.