Vertical Curves

Learning Objectives

  • Explain how a parabolic vertical curve joins two tangent grades.
  • Compute elevations, instantaneous grades, and an in-curve high or low point.
  • Distinguish crest sight-line control from sag headlight control.
  • Evaluate both sight-distance length domains and document the governing design basis.

Vertical curves connect changes in route grade. Their geometry supports profile continuity, while sight distance and other project criteria set the required length.

Why a vertical curve is used

A vertical alignment contains tangent grades joined by curves. A parabolic curve changes grade at a constant rate along horizontal distance, giving a direct elevation and grade at each station. The design still has to satisfy the sight-distance, comfort, drainage, appearance, and fit criteria adopted for the project.

Control points and grades on a vertical curve

The figure defines the PVC, PVI, PVT, curve length, and grade direction for a symmetrical curve. The zero-grade point is conditional; many curves have no internal high or low point.

A parabolic route profile joins the incoming and outgoing tangent grades at the PVC and PVT; the tangent grades meet at the PVI, and a zero-grade point is marked only when it falls inside the curve.

Profile notation

  • PVC (also BVC): point where the curve begins.
  • PVI: intersection of the two tangent grades.
  • PVT (also EVC): point where the curve ends.
  • G₁ and G₂: incoming and outgoing grades in percent.
  • L: horizontal curve length in metres.
  • ΔG = G₂ − G₁: signed grade change in percentage points; its sign sets the curve's concavity.
  • A = |ΔG|: absolute grade difference in percentage points, used for K and sight-distance relations.

Parabolic curve model

One consistent unit convention

In the equations below, use grades as percentage points and horizontal distances in metres. Divide grades by 100 when multiplying a grade by a distance to obtain an elevation change. Keep stationing and elevations in the project's own consistent units.

Elevation and grade along a symmetrical curve

Find profile elevation and instantaneous grade at horizontal distance x from the PVC.

y(x)=yPVC+G1100x+ΔG200Lx2y(x) = y_{PVC} + \frac{G_1}{100}x + \frac{\Delta G}{200L}x^2G(x)=G1+ΔGLxG(x) = G_1 + \frac{\Delta G}{L}x

Variables

SymbolDescriptionUnit
y(x)y(x)Profile elevation at distance xm
yPVCy_{PVC}Elevation at the PVCm
G1G_1Incoming tangent grade in percentage points%
ΔG\Delta GSigned change G_2 - G_1 in percentage points%
LLHorizontal curve lengthm
xxHorizontal distance from PVC, from 0 to Lm
G(x)G(x)Instantaneous grade at distance x%

Stationing and tangent elevations

For a symmetrical curve, the PVI lies halfway between the PVC and PVT in station. With the PVI elevation known, compute the PVC and PVT elevations by moving along their respective tangent grades:

StaPVC=StaPVI−L2,StaPVT=StaPVI+L2\mathrm{Sta}_{PVC}=\mathrm{Sta}_{PVI}-\frac{L}{2},\qquad \mathrm{Sta}_{PVT}=\mathrm{Sta}_{PVI}+\frac{L}{2}yPVC=yPVI−G1100L2,yPVT=yPVI+G2100L2y_{PVC}=y_{PVI}-\frac{G_1}{100}\frac{L}{2},\qquad y_{PVT}=y_{PVI}+\frac{G_2}{100}\frac{L}{2}

These midpoint relations do not apply to an asymmetric layout. Confirm the curve definition and station convention in the project records before setting out points.

K-value and the zero-grade point

Rate of vertical curvature

Relates curve length to the magnitude of grade change.

K=LA,L=KAK=\frac{L}{A},\qquad L=KA

Variables

SymbolDescriptionUnit
KKHorizontal curve length per one percentage point of grade changem/%
LLHorizontal curve lengthm
AAAbsolute grade difference |G_2-G_1|%

Find the high or low point only when it lies on the curve

Set the grade equation to zero. For a nonzero grade change, the candidate location is

x0=−G1LΔGx_0=-\frac{G_1L}{\Delta G}

It is an internal high or low point only if 0≤x0≤L0 \leq x_0 \leq L. Otherwise the curve has no horizontal tangent in its interval. A positive ΔG\Delta G gives a concave-up profile; a negative ΔG\Delta G gives a concave-down profile.

Crest and sag curves

Different geometry, different sight-distance control

A crest curve is convex. Its roadway profile can block a driver's line of sight, so a crest sight-distance model relates the curve to an eye point and an object point. A sag curve is concave. For the nighttime headlight model, available sight distance depends on the headlight height and beam angle. The governing agency and project standard provide the applicable input assumptions and design criteria.

Sight-distance mechanisms on crest and sag curves

The two panels compare the physical mechanisms behind crest and sag sight-distance checks. Eye, object, headlight, and beam inputs must come from the adopted design basis.

The crest profile hides an object beyond the curve from an eye point, while the sag profile directs a headlight beam toward a roadway point ahead.

Sight-distance length equations

The following SI forms use AA in percentage points, SS and LL in metres, heights in metres, and the headlight angle β\beta in degrees when evaluated by a calculator. For crest curves, h1h_1 is the adopted eye height and h2h_2 is the adopted object height. For sag curves, hh is the adopted headlight height. These are geometric models; use input values and any additional criteria required by the current jurisdictional standard.

Crest curve sight distance

Candidate curve length for the eye-to-object sight-line model.

L=AS2200(h1+h2)2(S≤L)L=\frac{A S^2}{200(\sqrt{h_1}+\sqrt{h_2})^2}\quad(S\leq L)L=2S−200(h1+h2)2A(S≥L)L=2S-\frac{200(\sqrt{h_1}+\sqrt{h_2})^2}{A}\quad(S\geq L)

Variables

SymbolDescriptionUnit
LLCandidate minimum curve length; verify the stated domainm
SSRequired sight distance from the adopted design basism
AAAbsolute algebraic grade difference in percentage points%
h1h_1Adopted driver's eye heightm
h2h_2Adopted object heightm

Sag curve headlight sight distance

Candidate curve length for the headlight sight-line model.

L=AS2200(h+Stan⁡β)(S≤L)L=\frac{A S^2}{200(h+S\tan\beta)}\quad(S\leq L)L=2S−200(h+Stan⁡β)A(S≥L)L=2S-\frac{200(h+S\tan\beta)}{A}\quad(S\geq L)

Variables

SymbolDescriptionUnit
LLCandidate minimum curve length; verify the stated domainm
SSRequired sight distance from the adopted design basism
AAAbsolute algebraic grade difference in percentage points%
hhAdopted headlight heightm
β\betaAdopted upward headlight-beam angledeg
Choose the length equation that matches S and L

The length equations have separate geometric domains. Compute a candidate, check whether its assumed relation between S and L holds, and recompute with the other form if needed.

Two schematic profiles distinguish the case where sight distance S is no longer than curve length L from the case where S extends beyond L.

Treat sight distance as one design check

The equations return a length candidate under their assumptions. Check the candidate's S/LS/L domain, then evaluate all other criteria required by the project, such as drainage on a sag curve or comfort and appearance controls. The selected length must satisfy the controlling requirement and fit the alignment. Published tables and input values may be jurisdiction- and edition-specific; do not transfer a value without checking the governing standard.

Length-selection workflow

Apply the adopted project criteria

The chart includes the crest or sag model, both sight-distance domains, other length controls, and a fit check. Record the source of the design speed, sight distance, geometric inputs, and selected criteria with the calculation.

Select and verify vertical-curve length

Crest and sag curves use different sight-distance geometry. Use the design speed, sight-distance requirement, vehicle assumptions, and minimum criteria from the governing project standard.

Evaluate the S≤L and S≥L equations for each applicable criterion and retain a candidate whose result satisfies its assumed domain. Other project requirements, including comfort, drainage, or appearance criteria, may govern the selected length.

Select and verify vertical-curve lengthCrest and sag curves use different sight-distance geometry. Use the design speed, sight-distance requirement, vehicle assumptions, and minimum criteria from the governing project standard.. Evaluate the S≤L and S≥L equations for each applicable criterion and retain a candidate whose result satisfies its assumed domain. Other project requirements, including comfort, drainage, or appearance criteria, may govern the selected length.. Start → Set governing standard, design speed, SSD, and sight inputs; Set governing standard, design speed, SSD, and sight inputs → Record G₁ and G₂; compute A = |G₂ − G₁|; Record G₁ and G₂; compute A = |G₂ − G₁| → A = |G₂ − G₁| > 0?; A = |G₂ − G₁| > 0? — Yes → Crest curve?; A = |G₂ − G₁| > 0? — No → Hold: A = 0; verify grade data; Crest curve? — Yes → Use crest eye-to-object sight-distance model; Crest curve? — No → Use sag headlight sight-distance model; Use crest eye-to-object sight-distance model → Evaluate S≤L and S≥L candidate equations; Use sag headlight sight-distance model → Evaluate S≤L and S≥L candidate equations; Evaluate S≤L and S≥L candidate equations → Candidate satisfies its S/L domain?; Candidate satisfies its S/L domain? — Yes → Review comfort, drainage, appearance, and project controls; Candidate satisfies its S/L domain? — No → Recompute using the alternate domain equation; Recompute using the alternate domain equation → Evaluate S≤L and S≥L candidate equations; Review comfort, drainage, appearance, and project controls → Select the greatest applicable required length; Select the greatest applicable required length → Length meets criteria and fits the alignment?; Length meets criteria and fits the alignment? — Yes → Set stations and elevations; document the design basis; Length meets criteria and fits the alignment? — No → Revise profile or criteria inputs; recalculate; Revise profile or criteria inputs; recalculate → Set governing standard, design speed, SSD, and sight inputs; Set stations and elevations; document the design basis → End

Start → Set governing standard, design speed, SSD, and sight inputs; Set governing standard, design speed, SSD, and sight inputs → Record G₁ and G₂; compute A = |G₂ − G₁|; Record G₁ and G₂; compute A = |G₂ − G₁| → A = |G₂ − G₁| > 0?; A = |G₂ − G₁| > 0? — Yes → Crest curve?; A = |G₂ − G₁| > 0? — No → Hold: A = 0; verify grade data; Crest curve? — Yes → Use crest eye-to-object sight-distance model; Crest curve? — No → Use sag headlight sight-distance model; Use crest eye-to-object sight-distance model → Evaluate S≤L and S≥L candidate equations; Use sag headlight sight-distance model → Evaluate S≤L and S≥L candidate equations; Evaluate S≤L and S≥L candidate equations → Candidate satisfies its S/L domain?; Candidate satisfies its S/L domain? — Yes → Review comfort, drainage, appearance, and project controls; Candidate satisfies its S/L domain? — No → Recompute using the alternate domain equation; Recompute using the alternate domain equation → Evaluate S≤L and S≥L candidate equations; Review comfort, drainage, appearance, and project controls → Select the greatest applicable required length; Select the greatest applicable required length → Length meets criteria and fits the alignment?; Length meets criteria and fits the alignment? — Yes → Set stations and elevations; document the design basis; Length meets criteria and fits the alignment? — No → Revise profile or criteria inputs; recalculate; Revise profile or criteria inputs; recalculate → Set governing standard, design speed, SSD, and sight inputs; Set stations and elevations; document the design basis → End

  • Start: terminator
  • Set governing standard, design speed, SSD, and sight inputs: input output
  • Record G₁ and G₂; compute A = |G₂ − G₁|: process
  • A = |G₂ − G₁| > 0?: decision
  • Hold: A = 0; verify grade data: terminator
  • Crest curve?: decision
  • Use crest eye-to-object sight-distance model: process
  • Use sag headlight sight-distance model: process
  • Evaluate S≤L and S≥L candidate equations: process
  • Candidate satisfies its S/L domain?: decision
  • Recompute using the alternate domain equation: process
  • Review comfort, drainage, appearance, and project controls: process
  • Select the greatest applicable required length: process
  • Length meets criteria and fits the alignment?: decision
  • Revise profile or criteria inputs; recalculate: process
  • Set stations and elevations; document the design basis: process
  • End: terminator

Asymmetric curves

Check continuity at every curve junction

An asymmetric arrangement may be used when equal tangent lengths do not fit the alignment. Define its geometry explicitly. If it is modeled as adjoining parabolic arcs, their elevations and grades must agree at the shared junction so the profile remains continuous. Use the equations and design checks for that specific curve definition; the symmetrical midpoint station and elevation relations above cannot simply be reused.

Interactive profile staking

Explore geometry with the profile simulation

Adjust the incoming and outgoing grades, curve length, roadway width, and staking interval. Inspect PVC, PVI, PVT, calculated elevations, instantaneous grade, and whether a zero-grade point lies inside the curve. The roadway view uses one physical scale in every direction; its fallback profile graph is fitted to the displayed data.

Symmetrical Vertical Curve — 3D Profile & Field Staking

Concept and model scope

Connect incoming and outgoing grades to the true-scale parabolic roadway, then stake elevations and verify that the terminal grade at PVT closes to g₂.

The roadway uses one physical scale in longitudinal, vertical, and transverse directions; no vertical exaggeration is applied in the WebGL scene. The model is a symmetrical parabolic vertical curve with a 100.000 m PVC datum.

Domains: grades −8% to +8%, curve length 80–400 m, roadway width 5.5–12 m, and stake interval 10–40 m. The static fallback is a profile graph whose vertical axis is independently fitted for readability.

Preparing profile scene…
Field sequence

Use the three-step sequence to connect tangent grades, the parabolic roadway, station staking, and PVT grade closure.

Controls

Geometry changes clear recorded observations.
4.00%
-3.00%
180 m
7.2 m
20 m
Active profile station1 / 10
Curve geometry
Type
Crest curve
A = |g₂ − g₁|
7.00%
K = L/A
25.71 m/%
PVI offset E
1.575 m
PVT elevation
100.900 m
High point
102.86 m / 102.057 m
Active station observation
Distance from PVC
0.00 m
Elevation
100.000 m
Instantaneous grade
4.000%
Field record
Not recorded
Field record: 0 of 10 stations captured

PVT must finish at g₂ = -3.000%. Computed terminal-grade closure error: 0.000000%.

Key Takeaways
  • A parabolic vertical curve changes grade at a constant rate along horizontal distance.
  • Keep percentage grades and horizontal distances consistent; divide grade by 100 in elevation changes.
  • K=L/AK=L/A describes curve length per percentage point of grade change.
  • A zero-grade point is on the curve only when its computed distance satisfies 0≤x0≤L0\leq x_0\leq L.
  • Crest and sag sight-distance models describe different geometric mechanisms, and each has S≤LS\leq L and S≥LS\geq L cases.
  • Verify equation domains, project-specific inputs, other required criteria, and alignment fit before selecting a length.

References