Free · Surveying geometry

Symmetric Parabolic Vertical Curve

Find elevations and stations along a symmetric parabolic vertical curve from its beginning elevation, signed grades and horizontal length.

Formula & worked example
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How to use this calculator

Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

With BVC elevation 121 ft, initial grade +1.75%, final grade −1%, and horizontal curve length 1600 ft, the curve elevation 800 ft from the BVC is 129.5 ft. The intersecting tangents meet at elevation 135 ft; that PVI elevation is not the curve elevation at the midpoint.

z(x) = z₀ + g₁x/100 + (g₂−g₁)x²/(200L), for percent grades. The tangent PVI is at x=L/2 with elevation z₀+g₁L/200; the EVC is at z₀+(g₁+g₂)L/200.

Worked example

Enter these known values and leave the other values blank.

Beginning curve elevation (BVC): 121 m
Initial grade (%): 1.75 %
Final grade (%): -1 %
Horizontal curve length (L): 1600 m
Horizontal offset from BVC (x): 800 m
Beginning station: 11200 m

Tangent intersection elevation (PVI): 135 m
End curve elevation (EVC): 127 m
Tangent intersection station: 12000 m
End station: 12800 m
Curve elevation at x: 129.5 m

Assumptions and limitations

This is one symmetric quadratic vertical curve joining two straight tangent grades. Curve length means horizontal distance, not the distance along the sloped profile. Both halves have equal horizontal length.
Enter signed grades as percentage numbers: 2 means a rise of 2 units per 100 horizontal units, and −1 means a fall of 1 per 100. All elevations use the same datum. Negative datum elevations are meaningful.
The PVI is the intersection of the two tangent lines. Except when the grades are equal, it is not a point on the parabolic roadway profile. At the midpoint, the curve and PVI elevations differ by (g₂−g₁)L/800.
The point offset x is measured from the BVC, so x=0 gives the beginning elevation and x=L gives the end elevation. It is not an absolute station. The beginning station is independently added to obtain absolute PVI and EVC stations.
The captured workflow restricts the initial and final grades to opposite signs or zero. This is a source-specific subset: a parabola can connect same-sign grades, but that broader case is not admitted here.
A valid curve has strictly positive horizontal length and 0≤x≤L. A zero grade sum can make the length inverse indeterminate from endpoint elevations; do not infer a unique length from equal endpoint elevations alone.
The point elevation is a one-way result. The retained beginning/end/PVI elevation and station relationships can solve their original inverse targets when the required independent values are known. Clear the intended unknown before supplying other measurements.
The original profile has 101 equally spaced horizontal samples. Its chart rounds horizontal positions to four significant digits and elevations to three decimal places; a chart point does not replace the scalar calculation.
Legacy US survey feet and international feet are distinct. This isolated candidate replaces the captured rounded survey-foot factor with the exact 1200/3937 meters. NIST deprecated that unit after December 31, 2022 except for historical and legacy use. International feet remain exactly 0.3048 meters.
Numeric entries use the existing bounded exact finite-decimal arithmetic grammar, at most 40 significant result digits and nonzero exponent −1000 through 1000 before and after input unit conversion. Derived values can lie outside the re-entry range. Unsupported functions, symbols, repeating expressions and lossy arithmetic are refused with decimal guidance. These are computational limits, not physical curve-size limits.
A source-pinned scientific display fallback retains a nonzero scalar when the original relative rounding cannot display it within 1000 decimal places. Ordinary source rounding remains unchanged. The finite chart can become unavailable while a scalar remains valid.
This is geometric interpolation, not road-design approval. It does not check sight distance, drainage, comfort, minimum curve lengths, clearance, superelevation, construction tolerances or jurisdictional standards.

Common questions

Why does the midpoint elevation differ from the PVI?

The PVI belongs to the tangent lines. The parabolic curve transitions between those lines and normally passes above or below their intersection.

Is the distance of point x an absolute station?

It is the horizontal offset from the beginning of the curve. Add the beginning station if an absolute station is needed.

Does this calculate a safe minimum vertical-curve length?

It interpolates an already specified curve. It does not select a length from design speed, sight distance, comfort or local requirements.

References

Preserve all twelve original controls and sixteen inverse variants. Replace the nine legacy survey-foot selectors with exact 1200/3937 metre relationships; keep international feet distinct. Use exact-sign point and grade conditions, positive horizontal length, source-bound scientific display and three literal101-point chart translations with the captured approximate chart unit factors and explicit unavailable plotting states.

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