Vertical Curve Elevation Point Calculator

How to Use the Calculator

1. Enter the Distance \(x\) from the Point of Vertical Curvature (PVC) to the point where you need the elevation, in feet. It must be between 0 and the curve length.
2. Enter the Initial Grade \(g_1\) as a decimal: 3% becomes 0.03, and a downgrade is negative.
3. Enter the Final Grade \(g_2\) the same way, negative for downgrades.
4. Enter the Length of vertical curve \(L\) in feet. This is the horizontal length from PVC to PVT, not the arc length.
5. Enter the Initial Elevation epvc — the elevation of the roadway at the PVC, in feet.
6. Read the result: Elevation y is the elevation of the curve at your chosen distance, in feet.

What Is the Elevation Point of a Vertical Curve?

When a road or railway changes from one grade to another, the two tangent grades are joined by a vertical curve — almost always a parabola in modern design. The curve begins at the Point of Vertical Curvature (PVC) and ends at the Point of Vertical Tangency (PVT), with the Point of Vertical Intersection (PVI) where the two grade lines would meet. The parabolic shape gives a constant rate of change of grade, which produces the smooth, comfortable ride that a simple angular break cannot.

Because the curve deviates from the straight grade lines, the elevation at any station along the curve is not the same as the elevation of either tangent. The equation used by this calculator adds a parabolic correction term \(\frac{(g_2 - g_1)x^2}{2L}\) to the tangent elevation \(EPVC + g_1 x\). At \(x = 0\) the correction is zero (the curve matches the incoming grade), and at \(x = L\) the curve leaves on the outgoing grade \(g_2\), so the geometry closes perfectly at both ends.

The result tells a designer or contractor exactly how high the finished pavement, rail, or drainage invert should be at a given station. This is essential for setting construction stakes, checking earthwork quantities, verifying clearance over utilities and drainage, and confirming that headlight sight distance and stopping sight distance criteria are met on crest and sag curves.

The sign of the correction term follows the sign of \(g_2 - g_1\). On a crest curve (grade going from positive to negative, or decreasing), the correction is negative and the curve lies below the tangents. On a sag curve (grade increasing), the correction is positive and the curve lies above the tangents. In the worked example above, the curve is a crest, so the elevation 50 ft in is slightly lower than the extended tangent.

Assumptions, Limitations, and Practical Tips

• Grades must be decimals, not percentages. Enter 0.03 for +3% and -0.01 for -1%. Mixing percent values with feet will produce meaningless results.
• Distances are horizontal. Both \(x\) and \(L\) are measured horizontally, which is how vertical curves are defined in highway design practice.
• \(x\) should stay between 0 and \(L\). Inside that range the equation describes the actual curve. Outside it, the formula still computes a value, but it extrapolates the parabola beyond the real curve, which is not a valid design elevation.
• Assumes an equal-tangent, symmetric parabola. The PVC and PVT are assumed equidistant from the PVI. Unequal-tangent curves need a modified formula with different offsets on each side of the PVI.
• Units must be consistent. This calculator works with feet; if your project is metric, use meters for all lengths and elevations and the result will be in meters.
• To find the highest (or lowest) point of the curve, set the first derivative to zero: \(x_{high} = \frac{g_1 L}{g_1 - g_2}\) for a crest curve. This is where the grade passes through zero and is useful for checking drainage or sight lines.
• If you are also estimating material quantities for the project, our Earthwork Cross Sectional Area and Earthwork Cross Section Volume calculators can be used alongside this one to quantify cuts and fills, and the Concrete Yardage calculator helps with pavement or structure pours.