Vertical Design · Study · PE Civil: Transportation · FE → PE Prep
Vertical Design
11% of exam
Equal-tangent parabolic curves: elevations and offsets, high/low points, PVI/PVC/PVT stationing, and crest/sag curve lengths from stopping and passing sight distance and the rate of vertical curvature K.
5 concepts
A. Vertical alignment
Equal-Tangent Parabolic Curves
How the symmetrical parabola joins two grades — PVC/PVI/PVT stationing, the tangent-offset elevation model, and locating the high or low point.
A vertical curve is the smooth transition that carries a road from one grade onto another — from a +3% upgrade onto a −2% downgrade over a crest, or out of a sag. The PE Civil exam tests the geometry of that transition relentlessly, because every elevation on the road profile, every sight-distance check, and every drainage low point depends on it. Get the parabola model and the stationing arithmetic right and a whole family of problems falls open. The governing relations are compact enough to live on a single page of the NCEES PE Civil Reference Handbook (§5.3.1, Symmetrical Vertical Curve Formula), so the points are won not by memorizing more, but by applying that one model cleanly — tangent elevation plus offset, every time.
Why a parabola, not a circular arc
Highway vertical curves are second-degree parabolas, not circular arcs, for one practical reason: a parabola produces a constant rate of change of grade. If the grade changes uniformly with distance, the vertical acceleration a driver feels is constant and the curve is comfortable and easy to stake. The defining property is that the second derivative of elevation with respect to horizontal distance is a constant r, the rate of change of grade. Integrate that twice and you get the elevation profile — a quadratic in the distance x measured from the curve's start.
dx2d2Y=r=Lg2−g1=constant
The three control points and their stations
Every equal-tangent curve is anchored by three points. The PVI (point of vertical intersection) is where the two tangent grades, if extended, would meet — the vertex. The PVC (point of vertical curvature) begins the curve on the back tangent, and the PVT (point of vertical tangency) ends it on the forward tangent. Because the curve is symmetrical, the PVC and PVT sit one half-length to each side of the PVI. Station arithmetic is where careless points are lost: remember a station is 100ft, so subtract L/2 in feet from the PVI station and convert back to station form.
StaPVC=StaPVI−2L,StaPVT=StaPVI+2L
Elevations: tangent grade plus tangent offset
The cleanest way to compute any elevation on the curve is the handbook's tangent-offset model: first ride the back-tangent grade g1 from the PVC, then add the parabolic offset y that pulls the curve off that tangent. With x measured horizontally from the PVC, the curve elevation is Y=YPVC+g1x+y, where the offset grows with x2. Writing A=g2−g1 as the algebraic difference in grades (in percent) and L and x in feet, the offset takes the famous form below. The sign of A carries everything: A<0 is a crest (curve below the back tangent), A>0 is a sag.
y=200LAx2,Y=YPVC+g1x+200LAx2
The middle ordinate E and the offset symmetry
At the PVI itself (x=L/2) the tangent offset reaches its maximum magnitude, the external or middle ordinate E. Substituting x=L/2 into the offset gives E=∣A∣L/800 (with A in percent and L in feet); E is reported as a positive distance, and whether the curve lies below the tangent (crest) or above it (sag) is carried by the sign of A, not by E. A handy property follows: because offsets scale with the square of the distance from the nearest curve end, the offset at any fraction of the curve is that fraction squared times E — measured from whichever tangent is nearer. This lets you sketch and check a profile fast.
E=frac∣A∣,L800,qquadyx=Eleft(fracxL/2right)2
Locating the high or low point
The crest's high point (or sag's low point) is where the curve grade momentarily reaches zero — the turning point that controls drainage inlets, overpass clearance, and sight distance. Differentiate the elevation equation and set it to zero: the grade on the curve at distance x is g1+(A/100)(x/L), which vanishes at xm below. Note xm is measured from the PVC; if it comes out negative or larger than L, the grade never turns inside the curve (both grades have the same sign) and there is no interior high or low point.
xm=−Ag1L=g1−g2g1L
Exam strategy
Build a tiny table every time: list PVC station and elevation first (from the PVI by backing off g1 over L/2), then march x from the PVC. Compute Y=YPVC+g1x+(A/200L)x2 — never mix this with a forward-tangent reference mid-problem. Keep grades in percent when you use the 200L and 800 constants, and keep x, L in feet. For the high/low point, find xm first, confirm 0≤xm≤L, then feed it back into the same elevation equation. A fast sanity check: the offset magnitude at the PVI must equal E=∣A∣L/800, and the high-point offset magnitude exceeds E only when the turning point lies past the PVI (otherwise it is less).
Key equations
Rate of change of grader=Lg2−g1=LA
Constant for a parabola. g1,g2 = grades (decimal or %), L = curve length (ft or stations); A=g2−g1
Vertical distance from the back tangent to the curve. A in percent (its sign sets the direction:
PVC and PVT stationsStaPVC=StaPVI−2L,StaPVT=StaPVI+2L
PVC elevation from PVIYPVC=YPVI−g12L
Middle ordinate (external) at PVIE=frac∣A∣,L800
Maximum tangent offset, occurring at the PVI, reported as a positive magnitude. A in percent, L in ft, E
High / low point stationxm=−Ag1L=g1−g2g1L
Rate of vertical curvatureK=∣A∣L
Horizontal length per 1% change of grade (ft/%). The single parameter that links curve length to sight-distance design.
Grade on the curveg(x)=g1+LAx
Instantaneous grade (%) at distance
Worked examples
Elevation and high point on a crest curve
Problem. A crest vertical curve joins a g1=+3.00% back grade to a g2=−2.00% forward grade. The curve is L=600ft long and the PVI is at station 50+00 with elevation 1240.00ft. Find the PVC and PVT stations, the curve elevation at station 48+50, and the station and elevation of the high point.
Solution. Stations: StaPVC=50+00−3+00=47+00 and StaPVT=50+00+3+00=53+00
Sizing a sag curve to hold a low-point elevation
Problem. A sag vertical curve joins g1=−2.50% to g2=+1.50% with the PVI at station 70+00
Common pitfalls
•Stationing arithmetic: L/2 is in feet, but stations read in hundreds. Subtracting 300ft from station 50+00 gives 47+00, not 49+70. Convert deliberately.
•Sign of A: a crest has A=g2−g1<0 (offset pulls the curve down). Dropping the sign flips a crest into a sag and reverses the high/low point.
•Mixing grade units with the constants: the 200L and 800 forms expect A in percent. If you carry A as a decimal, the offset is off by 100.
•Referencing the wrong tangent mid-problem: compute every elevation from the PVC and the back tangent g1. Switching to the forward tangent halfway corrupts the offset.
•Assuming a high/low point always exists. If g1 and g2 have the same sign, xm
•Confusing E=AL/800 (offset at the PVI) with the high-point offset. They coincide only for a symmetric curve with ∣g1∣=∣g2∣
References
NCEES PE Civil Reference Handbook — §5.3.1 Symmetrical Vertical Curve Formula
AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book, GDHS-7) — Vertical Alignment — Profile design context and grade-control practice.
Unequal-Tangent (Asymmetric) Vertical Curves
When the two tangent legs are different lengths — two parabolas meeting at the CVC, the offset there, and elevations and the turning point on each branch.
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B. Stopping and passing sight distance
Crest Vertical Curves & Sight Distance
Sizing a crest curve so the pavement itself does not hide the road ahead — the S<L and S>L length equations, the K-rate, and AASHTO eye and object heights.
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Sag Vertical Curves
Why a sag curve is sized by headlight reach at night — plus the comfort, drainage, and overhead-clearance criteria and the K-value that governs.
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Sight Distance & Design K-Values
The braking-plus-reaction model for stopping and passing sight distance, the grade correction, and how design K = L/A ties sight distance to curve length.
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is the algebraic grade difference.
Elevation at horizontal distance x (ft) from the PVC. A in percent, g1 in decimal, L, x in ft. Use g1 as decimal here (e.g., +0.03).
A<0
crest, offset below the tangent;
A>0
sag, offset above),
x,L
in ft. The reported offset is the magnitude
∣y∣
; the sign only flags crest vs sag.
Symmetrical curve: half the length each side of the PVI. Convert L/2 (ft) to station form (1sta=100ft).
Back off the back-tangent grade over the half-length. Likewise YPVT=YPVI+g2L/2.
in ft. The sign of
A
(crest vs sag) sets which side of the tangent the offset falls on, not the size of
E
.
Distance from PVC to the turning point. Valid only if 0≤xm≤L; otherwise no interior high/low point. g1 and A same units.
.
Sanity check (offsets compared as magnitudes; the sign of
A
only marks this as a crest):
E=∣A∣L/800=5.00(600)/800=3.75ft
is the curve-to-tangent offset at the PVI (
x=L/2=300ft
). The back-tangent offset magnitude
∣y∣=∣(A/200L)x2∣
keeps growing past the PVI, so at the high point (
x=360ft>300ft
) the offset is
∣−5.40∣=5.40ft
, larger than
E=3.75ft
— exactly what you expect for a point between the PVI and the PVT. Grade check:
g(360)=3.00+(−5.00/600)(360)=3.00−3.00=0
✓, confirming the turning point.
, elevation
842.00ft
. To keep minimum cover over a buried storm drain, the curve's low point must sit at exactly
845.00ft
. What curve length
L
achieves that?
Solution. Let L be unknown. A=g2−g1=1.50−(−2.50)=+4.00%.
PVC elevation: YPVC=YPVI−g12L=842.00−(−0.025)2L=842.00+0.0125L.
Low point from PVC: xm=−Ag1L=−0.04(−0.025)L=0.625L (inside [0,L] ✓).
Low-point elevation, in terms of L:
Ylp=YPVC+g1xm+200LAxm2=(842.00+0.0125L)+(−0.025)(0.625L)+200L4.00(0.625L)2.
The three L-terms combine cleanly: 0.0125L−0.015625L+0.0078125L=0.0046875L, so Ylp=842.00+0.0046875L.
Set Ylp=845.00: 0.0046875L=3.00⇒L=0.00468753.00=640ft.
Final: L=640ft.
Verify: YPVC=842.00+0.0125(640)=850.00ft; xm=0.625(640)=400ft; Ylp=850.00−0.025(400)+200(640)4.00(400)2=850.00−10.00+5.00=845.00ft ✓. Sanity check: Ylp rises monotonically with L (0.0046875ft/ft), so a longer sag dips less and lifts the low point — to raise the low point you lengthen the curve, the direction the algebra confirms.
falls outside
[0,L]
and the road keeps climbing or falling through the curve.