GELE Surveying (Geomatics) — Vertical (Parabolic) CurvesCheat Sheet
Vertical (Parabolic) Curves cheat sheet — the reference card you wish you had on exam day. Condensed from the full study notes, this is the high-yield core of Vertical (Parabolic) Curves for GELE Surveying (Geomatics). Download, print, revise.
Exam context
The Geodetic Engineer Licensure Examination is conducted by Professional Regulation Commission (PRC) — Board of Geodetic Engineering and is scheduled for September 2026. The Surveying (Geomatics) subtest is marked as "Core" in the official pattern, and Vertical (Parabolic) Curves appears in position 7th of 9 in the GELE Surveying (Geomatics) review rotation. Passing mark: 70% weighted average, no sub-test below 50%. Recent GELE 2026 papers have drawn roughly a meaningful share of questions from this subject.
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.
Sections
Formulas
Formula
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.
Formula
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).
Section Title
Fundamental Parabolic Curve Equation
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
Term
Vertical Parabolic Curve
Example
A +3% grade smoothly transitions to −2% grade over 200 m via a parabolic curve.
Definition
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.
Term
Rate of Grade Change (r)
Example
r = (−2% − 3%) / 200 m = −0.0005/m = −0.05%/m
Definition
The change in grade per unit length of curve; constant for a parabola, measured in m⁻¹ or %/m.
Term
Crest Curve
Example
+2% → −3%; sight line blocked by hump; limited by stopping distance.
Definition
A vertical curve where g₁ > g₂ (upgrade to downgrade); forms a summit; common sight-distance concern.
Term
Sag Curve
Example
−4% → +1.5%; night visibility and water drainage are design concerns.
Definition
A vertical curve where g₁ < g₂ (downgrade to upgrade); forms a valley; limited by headlight throw and drainage.
Term
PC (Point of Curvature) / BVC (Beginning of Vertical Curve)
Example
Station 10+000 m, elevation 100.00 m, grade +3%.
Definition
Start of the vertical curve; where the initial tangent grade g₁ meets the parabola.
Term
PT (Point of Tangency) / EVC (End of Vertical Curve)
Example
Station 10+200 m (after 200 m curve length), elevation given by formula.
Definition
End of the vertical curve; where the parabola meets the final tangent grade g₂.
Term
PI (Point of Intersection)
Example
For PC at 10+000 and L = 200 m, PI is at 10+100 m.
Definition
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.
Formulas
Formula
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.
Section Title
Turning Point (Summit or Low Point) Location
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
Term
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.
Definition
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
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.
Equation
dy/dx = g₁ + r·x = 0 ⟹ x = −g₁/r = (g₁·L) / (g₁ − g₂)
Conditions
At the turning point, the grade (slope) is zero.
Formulas
Formula
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).
Formula
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.
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.
Section Title
Elevation Calculations Along the Curve
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
Term
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.
Definition
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).
Formulas
Formula
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.
Formula
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.
Section Title
Vertical Offset and Mid-Curve Properties
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
Term
Maximum Vertical Offset
Example
For L = 200 m and A = 5%, offset_max = 5 × 200 / 800 = 1.25 m.
Definition
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.
Formulas
Formula
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.
Formula
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).
Formula
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
Section Title
Sight Distance and Vertical Curve Length
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
Term
Stopping Sight Distance (SSD)
Example
At 60 km/h, typical SSD ≈ 120 m (per NSCP and local standards).
Definition
The minimum distance a vehicle can travel and stop safely upon sighting an obstruction, including reaction time and braking distance.
Term
Crest Curve Sight-Distance Requirement
Example
A 5% crest curve with SSD = 120 m requires L ≥ 54 m (approx.).
Definition
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.
Term
Sag Curve Sight-Distance Requirement (Headlight)
Example
A 5% sag curve with SSD = 120 m requires L ≥ 90 m (approx.) for night visibility.
Definition
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.
Section Title
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
Term
Positive Grade (+)
Example
+4% grade on an approach to a bridge.
Definition
Upward slope in the direction of increasing station; +3% means 3 m rise per 100 m horizontal.
Term
Negative Grade (−)
Example
−3% grade leaving a hill.
Definition
Downward slope in the direction of increasing station; −2% means 2 m drop per 100 m horizontal.
Term
Crest Vertical Curve
Example
Hilltop intersection; line-of-sight limited.
Definition
Formed when initial grade g₁ > final grade g₂ (e.g., +3% to −2%); creates a summit; r < 0.
Term
Sag Vertical Curve
Example
Valley crossing; drainage and headlight visibility matter.
Definition
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.
Section Title
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
Term
Station
Example
PC at Sta. 5+200, PT at Sta. 5+400 (curve length L = 200 m).
Definition
Horizontal distance along the roadway centerline, typically measured from a fixed origin (e.g., 0+000, 10+500, where 10+500 = 10,500 m).
Term
Station–Elevation Table
Example
Sta. | Elev. (m) 5+200 | 100.00 5+220 | 100.54 5+240 | 101.04
Definition
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
Rows
Values
- g₁ > g₂
- g₁ < g₂
Property
Grade Relationship
Values
- r < 0 (negative)
- r > 0 (positive)
Property
Rate of Grade Change (r)
Values
- Concave down; summit (high point)
- Concave up; valley (low point)
Property
Curve Shape
Values
- Curve lies below initial tangent (offset negative)
- Curve lies above initial tangent (offset positive)
Property
Parabola vs. Tangent
Values
- Line of sight blocked by hump; limited by eye and object heights; day visibility
- Headlight throw limited; night visibility; also consider drainage
Property
Primary Sight-Distance Issue
Values
- Stopping sight distance (SSD); L = A·S² / (200·(√h₁ + √h₂)²)
- Headlight distance (night); L = A·S² / (400·h); also comfort and drainage
Property
Typical Design Control
Values
- Summit (maximum elevation); dy/dx = 0
- Low point (minimum elevation); dy/dx = 0
Property
Turning Point (if within curve)
Values
- +3% to −2% over 200 m
- −4% to +1.5% over 160 m
Property
Example
Columns
- Property
- Crest Curve
- Sag Curve
Table Title
Crest vs. Sag Vertical Curves
Rows
Values
- 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
Property
Find elevation at distance x from PC
Values
- x = (g₁·L) / (g₁ − g₂)
- g₁, g₂ (decimals), L
- Reversing denominator to (g₂ − g₁); forgetting to check 0 < x < L
Property
Find location of summit or low point
Values
- 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
Property
Find elevation at PT
Values
- 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)
Property
Find maximum vertical offset
Values
- 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
Property
Find minimum curve length (crest, SSD)
Values
- 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
Property
Find minimum curve length (sag, headlight)
Values
- r = (g₂ − g₁) / L
- g₁, g₂ (decimals), L
- Forgetting signs of g₁ and g₂; order matters (g₂ first in numerator)
Property
Find rate of grade change
Columns
- Question Type
- Formula(s) to Use
- Key Input
- Common Pitfall
Table Title
Vertical Curve Formula Selection Guide
Rows
Values
- 1.08
- m
- NSCP 2015; used in crest SSD formula
Property
Driver's Eye Height
Values
- 0.6
- m
- NSCP 2015; used in crest SSD formula
Property
Object Height (Passenger Car)
Values
- 0.6
- m
- NSCP 2015; used in sag headlight formula
Property
Headlight Height (Effective)
Values
- ~1°
- above horizontal
- Typical; affects sag curve design
Property
Headlight Beam Angle
Values
- 120
- m
- NSCP 2015 / PRC exam typical
Property
Min. Stopping SSD @ 60 km/h
Values
- 160
- m
- NSCP 2015 / PRC exam typical
Property
Min. Stopping SSD @ 80 km/h
Values
- 200
- m
- NSCP 2015 / PRC exam typical
Property
Min. Stopping SSD @ 100 km/h
Columns
- Parameter
- Standard Value
- Units
- Notes
Table Title
Common Exam Values and Standards
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