The hook
Vertical curves are parabolas connecting two roadway grades. Crests over hills, sags through valleys. The math is friendlier than horizontal curves — but the design driver is sight distance, not geometry. Slide g₁ positive and g₂ negative below to see a crest, then flip them to see a sag.
A (algebraic diff)-5.0%g₂ − g₁
K (rate)80.0L / |A|
Curve typeCrest
Mid-curve offset-2.50 ftA·L / 800
High/low x240 ft from PVC−g₁·L / A
Try this: set g₁ positive and g₂ negative — that's a crest curve, and the high point is somewhere inside it. Now flip the signs — the same shape becomes a sag with a low point. K-value (L / |A|) is what state DOTs publish in their road design manuals as a minimum for sight distance.
Memorize these
Concepts that show up on the exam
PVC / PVI / PVT
Point of Vertical Curvature (where the parabola begins), Point of Vertical Intersection (where the two grades would meet), Point of Vertical Tangency (where the parabola ends).
g₁, g₂ (Grades)
Incoming and outgoing grades, expressed as % (e.g., +3% means 3 ft rise per 100 ft run).
A (Algebraic difference)
A = g₂ − g₁. Negative for crest curves, positive for sag.
L (Length)
Horizontal distance from PVC to PVT. Standard practice: parabolic curves are SYMMETRICAL — equal distance L/2 each side of the PVI.
K (Rate of curvature)
K = L / |A|. The horizontal distance per 1% change in grade. State design manuals publish minimum K for each design speed (sight-distance driven).
High / low point
Only exists if g₁ and g₂ have opposite signs (crest with positive→negative, sag with negative→positive). Located at x = −g₁·L / A from the PVC.
Keep these in muscle memory
Formulas to know cold
Elevation along curve
y(x) = g₁·x + (A / 2L)·x²x is distance from PVC; grades in decimal form.
Algebraic difference
A = g₂ − g₁ (in % or decimal)Rate of curvature
K = L / |A| (ft per % of grade change)Mid-curve offset
M = A · L / 8 (in feet if grades are decimal)Often expressed as A·L/800 when grades are in %.
High / low point distance
x_max = −g₁ · L / A (only valid if g₁·g₂ < 0)Try it before you peek
Worked example
The problem
A crest curve has g₁ = +3.0%, g₂ = −2.0%, L = 400 ft. PVC elevation = 850.00 ft. Find: (a) K, (b) station distance of high point from PVC, (c) elevation of the high point.
Don't fall for these
What trips people up
Sign of A determines crest or sag
If A < 0 it's a crest. If A > 0 it's a sag. The mid-curve offset M takes A's sign — for a crest the curve sags BELOW the tangent intersection.
Grades as % vs. decimal
Pick one and stick with it. The formula
y = g₁·x + (A/2L)·x² requires consistent units. If grades are in %, divide by 100 in the formula.No high/low point when g₁·g₂ ≥ 0
If both grades are positive (or both negative), the curve is monotonic — no high/low point exists inside it. The formula will give you a value outside [0, L]; ignore it.
Test yourself
How well did it stick?
A quick 5-question check on Vertical Curves. See where you stand and what to review.