Unequal-tangent vertical curve
A vertical curve with different tangent lengths each side of the : , , and the point under the PVI, the high or low point, and the elevation at any station.
Experimental: not yet fully verified. How results are checked
The curve runs PVC 8+00.00 to PVT 14+00.00. At the PVI station it is at 96.667 ft, and the PVI is at 100 ft; the high point is on the curve.
- PVC elevation
- PVT station
- PVT elevation
- Curve elevation under the PVI
- Grade under the PVI
- High or low point station
Provenance
- Computed by
- survey.curves.unequal-vertical-curve 1.0.1, core 0.1.0
- Model
- Two parabolas, PVC to the point under the PVI and on to the PVT, sharing the grade g_c = (L1 g1 + L2 g2)/(L1 + L2) where they meet; the curve passes L1·L2·(g2 − g1)/(200(L1 + L2)) from the PVI (Ghilani & Wolf 2021, ch. 25)
- Accuracy
- Exact for the parabolas
- Notes
- 1 shown with the answer
- Cites
- Ghilani, C. D., and Wolf, P. R., Pearson, Elementary Surveying: An Introduction to Geomatics
Something look off?
How we got thisFormula, worked example, sources, and proof
Model: Two parabolas, PVC to the point under the PVI and on to the PVT, sharing the grade g_c = (L1 g1 + L2 g2)/(L1 + L2) where they meet; the curve passes L1·L2·(g2 − g1)/(200(L1 + L2)) from the PVI (Ghilani & Wolf 2021, ch. 25)
Accuracy: Exact for the parabolas
Worked example: g1 = +2%, g2 = −3%, 200 ft before and 400 ft after a PVI at 10+00. Source: Two-parabola unequal-tangent curve.
You enter
- Entering grade g1
- 2
- Leaving grade g2
- -3
- Length before the PVI
- 200 ft
- Length after the PVI
- 400 ft
- PVI elevation
- 100 ft
- PVI station
- 10+00
You get
- PVC station
- 8+00.00
- PVC elevation
- 96 ft
- PVT station
- 14+00.00
- PVT elevation
- 88 ft
- Curve elevation under the PVI
- 96.667 ft
- Grade under the PVI
- -1.3333
- High or low point station
- 9+20.00
- High or low point elevation
- 97.2 ft
- High or low point
- the high point is on the curve
Review: Not yet independently reviewed by a licensed surveyor.
Last verified: 2026-09-23, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.
Status: version 1.0.1, core 0.1.0. See this tool in the verification report.
Changes
- 2026-09-23, fixed: The unequal-tangent vertical curve's sentence said the curve passed, for example, "96.25 ft under the PVI", when 96.25 ft is the curve's elevation at the PVI station and the curve is 3.75 ft below the PVI. It now says "At the PVI station it is at 96.25 ft, and the PVI is at 100 ft". Version 1.0.1; no numbers change. Changelog
- 2026-09-23, fixed: Survey tools cite Ghilani and Wolf's Elementary Surveying, 16th edition (2021), but their method notes said "Ghilani & Wolf 2018". The notes now give 2021, the edition cited. No results change. Changelog
Checked against: 7 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. Chapter 25 (vertical curves: unequal-tangent curves).
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
Experimental means this tool has not yet met the stable bar: at least 20 golden vectors, differential tests, and an independent worked example.
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
Learn the concept: Vertical curves explained