The vertical curve calculator above works out the geometry of an equal-tangent parabolic vertical curve, the shape used to join two grades on a road or rail profile. From the incoming grade, the outgoing grade, the curve length and the station and elevation of the point of vertical intersection, it returns the PVC and PVT, the elevation at any station you name, the K value and the location of the high or low point.
Arb Digital builds free tools that compute what a formula genuinely gives and stop there. This page follows the parabolic vertical curve geometry set out in the American Association of State Highway and Transportation Officials' A Policy on Geometric Design of Highways and Streets, the document universally known as the Green Book. It publishes no K-value table and no sight-distance table, because those depend on design speed, curve type, driver eye and object heights and the criteria your agency has adopted. This is a geometry and teaching tool. A licensed civil engineer designs any real vertical alignment, and the road authority with jurisdiction sets the criteria it has to satisfy.
What This Vertical Curve Calculator Does
A vertical curve is a parabola, not a circular arc. The parabola is chosen because its rate of change of grade is constant, which produces a steady, predictable acceleration in the vertical plane and makes the arithmetic simple enough to lay out by hand. This tool implements the equal-tangent case, where the curve length is split evenly either side of the PVI.
Four groups of output come back. The tangent points, PVC at the start and PVT at the end, with their stations and elevations. The elevation at any station you enter, which is what a grading crew actually needs. The K value, which is the curve length per unit of algebraic grade difference and the parameter almost all design criteria are expressed in. And the turning point, the station where the curve is momentarily level, which is where drainage either works or does not.
The Federal Highway Administration's page on geometric design standards explains how state highway departments develop design standards through AASHTO, and that those standards form the basis of federal design requirements on the National Highway System. Which edition applies to a given project is a matter for the owning agency.
How to Use It
- Enter the two grades with their signs. Uphill in the direction of increasing stationing is positive. Getting a sign wrong flips a crest into a sag and moves the turning point off the curve entirely.
- Enter the curve length as a horizontal distance. Vertical curve length is measured along the horizontal projection, which is why a plan station difference is the right number to use.
- Give the PVI station and elevation. Stations are entered as plain distances, so 50+00 is 5000, and the display format only changes how the answers are written back.
- Name a station to report. Any value works. Outside the curve the tool extends the relevant tangent and labels the answer as tangent rather than curve.
- Supply the required K yourself. Take it from the sight-distance criteria your agency adopts for this design speed and curve type. The page compares it with the K you have; it never supplies one.
The Formula: How the Curve Is Calculated
Let A be the algebraic difference in grades, g2 minus g1, in per cent. The PVC is half the curve length back from the PVI, at elevation PVI elevation minus g1 over one hundred times half the length. The PVT is half the length forward, at PVI elevation plus g2 over one hundred times half the length.
For a distance x measured forward from the PVC, the elevation is the PVC elevation, plus g1 over one hundred times x, plus A divided by two hundred L, all times x squared. The K value is L divided by the absolute value of A. The turning point sits at x equal to minus g1 times L divided by A, and it only exists on the curve when that distance falls between zero and L, which happens when the two grades have opposite signs.
Work the defaults. Grades of plus three and minus two give A equal to minus five, so this is a crest curve, and K is 400 divided by 5, which is 80. The PVI is at station 5000 and elevation 1000, so the PVC is at 4800 with elevation 1000 minus 0.03 times 200, which is 994.00, and the PVT is at 5200 with elevation 1000 minus 0.02 times 200, which is 996.00. The turning point is at x equal to minus 3 times 400 divided by minus 5, or 240 from the PVC, which is station 5040. Its elevation is 994 plus 0.03 times 240, minus 0.0000625 times 240 squared, which is 994 plus 7.2 minus 3.6, or 997.60.
Why K Is the Number That Matters
Design criteria are almost never written as a curve length. They are written as a minimum K, because K carries the whole of the relationship between length and grade change in one figure, and sight distance depends on that ratio rather than on either quantity alone. Once you have a required K, the minimum length follows immediately as K times the absolute value of A.
K also has a direct physical reading: it is the horizontal distance required to change the grade by one per cent. A K of 80 means a hundred feet of curve for every 1.25 per cent of grade change. That makes it easy to sanity-check a profile at a glance, and it explains why a small grade change over a long curve produces an enormous K that is comfortably above any criterion.
The criteria themselves differ by curve type and by what is being checked. Crest curves are usually governed by stopping sight distance over the crest, with the driver eye height and object height the agency specifies. Sag curves are more often governed by headlight sight distance at night, and sometimes by comfort, drainage or overhead clearance under a structure instead. Because those criteria and their assumed heights vary between agencies and editions, this page takes the required K from you rather than choosing one; the current edition is A Policy on Geometric Design of Highways and Streets, 7th Edition, published by AASHTO. The stopping distance calculator covers the vehicle side of that question separately.
The Turning Point, Drainage and Flat Curves
The turning point is the one station on the curve where the grade is zero. On a crest it is the highest point; in a sag it is the lowest. It is not, except by coincidence, under the PVI: it sits under the PVI only when the two grades are equal and opposite.
Near that point the profile is almost flat for a considerable distance, and on a curbed road that is where water stops moving. A long, low-K sag curve can leave several hundred feet of gutter at less than half a per cent, which is why many agencies impose a separate drainage criterion on curbed sections independent of sight distance. The tool reports the turning point station and elevation precisely so that the flat region can be located, but whether the resulting drainage is acceptable is a design judgement, not an output.
When both grades have the same sign the turning point falls outside the curve and the profile climbs or falls throughout. The calculator says so rather than reporting a station that does not exist, which is the honest answer and also a useful check that the grade signs were entered correctly.
What This Calculator Deliberately Does Not Do
It does not handle unequal-tangent curves, where the two halves have different lengths and are treated as two parabolas meeting at a common point. It does not check clearance over or under a structure, which needs the obstruction geometry as well as the profile. It does not compute sight distance, because that depends on eye and object heights, and on whether the sight line stays on the curve, all of which are criteria rather than geometry.
Most importantly, it never reports that a curve is adequate. It reports the K you have, next to the K you told it you need, and the difference between them. Choosing the criterion, deciding which controls, and accepting the design are the responsibility of a licensed civil engineer working to the standards the road authority has adopted.
How This Page Sits Beside the Other Alignment Tools
The boundary in one sentence: this page fits a parabola between two known grades, while the elevation grade calculator works out a grade from a rise and a run in the first place. Use that one to establish g1 and g2, then bring them here.
The slope calculator covers slope as a mathematical ratio rather than a road grade, and the catenary curve calculator handles the quite different curve a hanging cable takes, which is often confused with a parabola but is not one. The parabola calculator gives the algebraic vertex form behind the geometry used here. For mixed-unit profiles the unit converter rescales lengths, and the rounding calculator keeps staked elevations reported to a consistent precision.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Dropping a grade sign — a plus three to minus two crest and a plus three to plus two continuous climb are different profiles entirely, and only the signs distinguish them.
- Measuring the length along the curve — vertical curve length is a horizontal distance, so it comes from the station difference, not from a sloped measurement.
- Assuming the high point is under the PVI — it only is when the grades are equal and opposite, and on an asymmetric grade change it can be well away from the midpoint.
- Treating a large K as automatically compliant — the governing criterion may be drainage, comfort or clearance rather than sight distance, and each has its own limit.
- Using this geometry as a design — a profile also has to satisfy the standards the road authority adopts, and only a licensed engineer can accept it.
Related Free Tools From Arb Digital
Establish the grades with the elevation grade calculator or the slope calculator, then bring them here. The parabola calculator covers the underlying algebra, the catenary curve calculator handles hanging-cable geometry, and the stopping distance calculator covers vehicle braking distance. Rescale mixed units with the unit converter, keep staking figures consistent with the rounding calculator, and browse the full free online tools hub for everything else.
Frequently Asked Questions
A parabola. It is used because the rate of change of grade is constant along it, which gives a steady vertical acceleration and makes offsets from the tangent easy to compute. Circular vertical curves are not standard practice in highway design.
Because minimum K depends on design speed, on whether the curve is a crest or a sag, on the sight-distance criterion being applied and on the eye and object heights the agency assumes. Those vary between authorities and between editions, so the required K is an input you supply from your own criteria.
At the station where the grade passes through zero, which is minus g1 times L divided by A, measured forward from the PVC. It coincides with the PVI only when the two grades are equal and opposite, and it does not exist on the curve at all when both grades share a sign.
One where the PVI sits exactly halfway between the PVC and the PVT, so each tangent is half the curve length. This tool handles that case only. Unequal-tangent curves are treated as two parabolas meeting at a common point and need a different computation.
Yes. Grades are percentages either way, and the arithmetic is unit-agnostic, so entering metres for length and elevation gives answers in metres. Stations can be displayed in hundreds or thousands to match the convention in use.
No. It gives geometry, not a design. A real profile has to satisfy sight distance, drainage, clearance, comfort and the standards the road authority has adopted, and it must be prepared and sealed by a licensed civil engineer.
Because the PVI is the intersection of the two tangents, not a point on the parabola. The curve passes below a crest PVI and above a sag PVI by the mid-ordinate, which is the algebraic grade difference times the length divided by eight hundred.
The algebraic difference is zero, there is no curve to compute, and K is undefined because it would require dividing by zero. The tool says so rather than printing an infinite value, since a zero grade change simply means the tangent continues.
This tool is provided for educational and preliminary geometric use only. It is not a road design, it is not sealed, and it does not evaluate sight distance, drainage, clearance or compliance with any standard. Design speed and every sight-distance criterion are inputs you supply. A licensed civil engineer prepares any real vertical alignment, and the road authority with jurisdiction governs the criteria that apply.
