Vertical Curve Calculator

Vertical curve calculator for roadway elevations

This vertical curve calculator estimates elevations along a simple parabolic vertical roadway curve. Enter the curve start station and elevation, curve length, entering grade, exiting grade, and station interval. The tool calculates PVC, PVI, and PVT stations, station-by-station elevations, the grade at each point, the curve's K value, and an interior high or low point when one falls within the curve.

Vertical curves connect two roadway grades smoothly in profile. A crest curve bends over a high point, while a sag curve bends through a low point. Surveyors, civil engineers, highway designers, students, and construction teams use vertical alignment calculations to understand how elevation changes along a road. This page provides a transparent geometry calculation for a symmetrical parabolic curve; it does not select a safe or compliant road design.

Results depend on the values entered and the stationing convention you use. Always check the project's design criteria, survey control, datum, units, agency requirements, sight distance, drainage, and design speed with a qualified transportation professional. The calculator is a planning and learning aid, not a stamped roadway design.

What is a vertical curve?

Road profiles contain tangent sections with different grades. Where one grade changes to another, a vertical curve provides a gradual transition. Instead of changing slope abruptly at the point where the tangents meet, the curve distributes the grade change over a selected horizontal length. A parabolic curve is widely used because its grade changes at a constant rate with horizontal distance.

A crest vertical curve is convex upward in a profile drawing: the road rises toward a crest and then may fall, or it may continue rising at a flatter grade. A sag vertical curve is concave upward: the road falls into a low area and then rises, or it may continue falling at a gentler grade. These names describe the direction of curvature, which depends on the difference between the outgoing and incoming grades.

The curve's beginning is commonly called the PVC, or point of vertical curvature. The tangent intersection is the PVI, or point of vertical intersection. The curve's end is the PVT, or point of vertical tangency. Some agencies or software use BVC and EVC instead of PVC and PVT. Confirm the notation used in your plans before comparing station values.

Inputs and how to use the calculator

Choose feet or metres first. Enter the PVC station as a distance along the project's baseline, such as 10+00 for station 1000 or 1+250 for station 1250. This calculator accepts the numeric distance value, not the plus-sign display format, so enter 1000 or 1250 respectively. The PVC elevation is the vertical value at the start of the curve, in the same length unit.

Enter the horizontal curve length, then the incoming and outgoing tangent grades as percentages. A 2 percent uphill grade is entered as 2; a 1.5 percent downhill grade is entered as -1.5. Positive grades rise with increasing station, and negative grades fall. The grade difference controls the curve's direction and rate of vertical change.

Set a station interval to create the elevation table. An interval of 25 feet produces points every 25 feet along the curve; a 10 metre interval produces points every 10 metres. PVC and PVT are included even when the interval does not land exactly on the endpoint. A smaller interval gives more rows but does not improve the quality of the underlying survey or design inputs.

Review the summary values, profile diagram, and station table together. The table can be exported as a CSV file for use in a spreadsheet or a compatible workflow. Before transferring data to design or staking software, confirm precision, units, coordinate system, station format, and project conventions.

Vertical curve formulas

The formula uses horizontal distance x measured forward from the PVC. Grades are converted from percent to decimal before calculating elevation. The algebraic grade difference A is outgoing grade minus incoming grade, in percentage points. The curve equation gives the vertical offset from the incoming tangent as the distance increases.

• A = g₂ − g₁, where grades are entered in percent
• PVT station = PVC station + curve length L
• PVI station = PVC station + L ÷ 2 (for a symmetrical curve)
• PVI elevation = PVC elevation + (g₁ ÷ 100) × (L ÷ 2)
• Elevation at distance x = PVC elevation + (g₁ ÷ 100)x + [A ÷ (200L)]x²
• Grade at distance x (%) = g₁ + (A × x ÷ L)
• K value = L ÷ |A|, with L in the selected length unit and A in percent
• Interior high/low point distance x = −g₁L ÷ A, when A is not zero and 0 < x < L

Worked example: finding elevations along a curve

Suppose a hypothetical curve begins at station 1000 with a PVC elevation of 100.00 feet. The curve is 400 feet long, the entering grade is +2.0 percent, and the exiting grade is -1.0 percent. The algebraic grade difference is -3.0 percentage points, so the profile is a crest curve.

The PVT station is 1000 + 400, or 1400. The PVI station is halfway along, at 1200. The tangent elevation at the PVI is 100 + 0.02 × 200, or 104.00 feet. This is the tangent intersection elevation; the parabolic curve elevation at that same station is lower by the vertical curve offset.

At x = 200 feet, the curve equation gives 100 + 0.02 × 200 + (-3 × 200² ÷ (200 × 400)) = 103.25 feet. The grade at that point is 2 + (-3 × 200 ÷ 400), or +0.5 percent. The high point occurs where the curve grade is zero: x = -(2 × 400) ÷ -3, approximately 266.67 feet from the PVC. These figures are only an example of the formula, not a road design recommendation.

Reading the station and elevation table

Each table row shows the station, the distance from PVC, the calculated elevation, and the instantaneous grade at that point. Station is the PVC station plus the relative distance. The elevation is calculated on the parabolic curve, rather than on either tangent line. Grade is the slope of the curve at that station and changes linearly between the entering and exiting values.

The selected interval determines the regular rows. The tool includes the PVC, interval stations, and PVT. If the interval does not divide the curve length evenly, the final row can be closer to the prior row than the selected interval. The curve's high or low point is calculated separately and may not coincide with a table row; use its reported station and elevation for that theoretical point.

The output is rounded for display. A rounded table is convenient for review but can hide small differences. Use appropriate precision for the project and retain unrounded source values where needed. CSV export preserves calculated numeric values at higher precision than the on-screen formatting, but it still needs independent checking before professional use.

Crest and sag curves, including the high or low point

The sign of A helps identify curve direction. If the outgoing grade is lower than the incoming grade, A is negative and the curve is a crest. If the outgoing grade is higher, A is positive and the curve is a sag. For example, changing from +3 percent to -2 percent creates a crest; changing from -3 percent to +2 percent creates a sag.

A high or low point exists inside the curve only when its grade reaches zero between PVC and PVT. The calculator solves the grade equation for zero and checks whether the resulting distance falls strictly within the curve length. A crest curve can have no interior high point if both tangent grades continue rising or both continue falling through the curve. The same is true for a sag curve and an interior low point.

The term high point or low point here is purely geometric. It does not assess visibility, drainage capacity, ponding, vehicle operation, pavement crossfall, or safety. A drainage low point, in particular, requires a broader design review than a vertical profile calculation.

Understanding K value and curve length

K is the horizontal curve length divided by the absolute algebraic grade difference expressed in percentage points. It describes how many units of horizontal distance are used for each one percentage-point change in grade. For a 400-foot curve with a 3-point grade difference, K is about 133.33 feet per percent. In metric units, a 120-metre curve with a 3-point change has K = 40 metres per percent.

K value is a useful way to compare vertical curve flatness, but it is not a universal design approval. Agencies may set minimum values based on design speed, stopping sight distance, headlight sight distance, comfort, drainage, terrain, and road classification. A larger K generally means a flatter transition for the same grade change, but the required value must come from the applicable design standards and project conditions.

Changing curve length while leaving grades fixed changes K and shifts the PVT station. It also changes where a theoretical zero-grade point falls. Do not choose a length solely to obtain a preferred K value without checking governing standards and constraints such as intersections, structures, access, property limits, and earthwork.

Stationing, units, and survey data

Roadway stationing is often displayed in plus format, where 10+00 represents 1,000 feet and 1+250 represents 1,250 metres. This calculator uses the continuous numeric distance, so remove the plus sign before entering a station. It reports station as a regular number; format it for plan sheets according to the project's station equation or agency convention.

Keep horizontal distance and vertical elevation in a consistent unit system. Feet-based inputs produce elevations and stations in feet; metre-based inputs produce metres. Grades remain percentages and do not change with the unit selection. When switching units, the calculator converts distances and elevations, while station intervals are converted as well.

Survey elevations must share the same vertical datum, benchmark basis, and project adjustment. A mathematically consistent profile can still be wrong if its starting elevation or station is wrong. Confirm the PVC with the approved alignment and profile data, and verify any station equation or alignment break before relying on the reported stations.

Common vertical curve calculation mistakes

One frequent error is entering grade as a decimal instead of a percent. Enter +2 for two percent, not 0.02. The calculator performs the percent-to-decimal conversion in the elevation equation. Another error is omitting a minus sign for a downhill grade, which changes the algebraic difference and the predicted curve shape.

Station values are sometimes entered in plus format or treated as an offset from zero. Enter the continuous station value and remember that the curve table adds distance to the PVC station. Also check whether a supplied length is horizontal curve length, not sloped distance along the pavement surface.

Do not confuse PVI elevation with the curve elevation at the PVI station. For a parabolic curve, the profile at the midpoint generally differs from the tangent intersection. Finally, make sure the vertical curve's endpoints and grades correspond to the same alignment and that all values use the same unit system.

Limits of this roadway profile calculator

This tool models one symmetrical parabolic vertical curve between two constant grades. It does not calculate a full roadway profile, horizontal geometry, cross sections, earthwork quantities, superelevation, sight distance, stopping distance, drainage, vehicle clearance, accessibility, or construction staking coordinates. It does not check design criteria or identify conflicts with intersections, utilities, bridges, or terrain.

The station table is based on the entered PVC station and elevation. The result does not validate the source survey, benchmark, geoid model, coordinate reference system, station equation, or alignment. The diagram is schematic and not a construction drawing. The tool cannot replace licensed professional engineering judgment, agency review, field verification, or approved project documents.

For an actual public road, private road, driveway, or site access design, use the current standards that apply to the location and road classification. Have a qualified civil or transportation engineer check the complete profile and all required safety and drainage criteria. Treat this calculator as an independent arithmetic aid only.

A practical review checklist

Before using a vertical curve estimate, confirm the intended PVC station and elevation, curve length, incoming grade, outgoing grade, and measurement units. Check that grades are signed correctly and entered as percentages. Review whether the curve is crest or sag and whether the reported high or low point lies inside its endpoints.

Compare the PVI and endpoint stations with the project profile. Spot-check at least one intermediate elevation by hand using the parabolic formula. Confirm that the station interval is suitable for the review, and inspect the CSV columns if you export a table. Keep a record of the input source and date if the values came from another design document.

For construction or public infrastructure decisions, have a qualified professional independently review the alignment and applicable criteria. This calculator can make the arithmetic visible, but it cannot determine whether a roadway profile is acceptable or safe.