Vertical curve

Model: Symmetric parabola: y(x) = y_PVC + g1·x + (g2 − g1)·x² / (2L)

Show your work

1. Algebraic difference

   A = g2 − g1, in percent

   -3% − 2% = -5%

2. Rate of change

   K = L / |A|, the length per percent of grade change

   600 ft / 5% = 120 ft per %

3. High point

   x = −g1 × L / A from the PVC, where the grade reaches zero

   −2% × 600 ft / -5%, from station 7+00.00 = 9+40.00

The same steps an agent gets from the MCP server with explain: true.

Accuracy: Exact

When to use this: Use this on a profile where one grade meets another: it gives the PVC and PVT stations and elevations, the high or low point, and the K value that design tables are read with, for a symmetric parabolic curve.

Limitations: It is the geometry of a symmetric curve, not a design: sight distance, drainage, and comfort criteria come from the governing design manual and its tables, which this does not reproduce. When both grades run the same way there is no high or low point within the curve, which the result says rather than inventing one.

Worked example: A crest curve: +2% to −3% over 600 ft, PVI 10+00 at 100.00 ft. Source: add-survey-suite scenario: PVC 7+00 at 94.00 ft; high point 9+40 at 96.40 ft; K = 120.

Review: Not yet independently reviewed by a licensed surveyor.

Last verified: 2026-09-20, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.0, core 0.1.0. See this tool in the verification report.

Changes

• 2026-09-19, result change: The horizontal curve, vertical curve, and area by coordinates are now stable, so their results no longer carry the EXPERIMENTAL_TOOL warning; the legacy-unit warning on area moved to the first position. They match FM 5-233 (Army Construction Surveying), the Indiana Design Manual vertical-curve example, and Wikipedia's shoelace example. No numbers changed. Changelog

Checked against: 21 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

• Elementary Surveying: An Introduction to Geomatics, Ghilani, C. D., and Wolf, P. R., Pearson, 16th edition. Chapters 10 (traverse computations), 12 (area), 24 (horizontal curves), 25 (vertical curves), 26 (volumes).
• A Policy on Geometric Design of Highways and Streets (the Green Book), American Association of State Highway and Transportation Officials, 7th edition. Table 3-34 (design controls for crest vertical curves) and Table 3-36 (sag), by design speed.

Terms

PVC: point of vertical curvature

On a vertical curve, where the curve leaves the incoming grade. Source: Elementary Surveying: An Introduction to Geomatics

PVI: point of vertical intersection

On a vertical curve, where the two grade lines would meet. Source: Elementary Surveying: An Introduction to Geomatics

PVT: point of vertical tangency

On a vertical curve, where the curve joins the outgoing grade. Source: Elementary Surveying: An Introduction to Geomatics