What's the K-factor for?

K = L / |A| where L is curve length and A is algebraic difference of grades (%). K is the horizontal distance per percent grade change. AASHTO publishes K_min by design speed for crest (sight distance) and sag (headlight) curves.

What's a typical K_min?

Crest, 55 mph: K = 114. Sag, 55 mph: K = 115. Sag K is larger than crest at low speeds, similar at high speeds. AASHTO Tables 3-34 (crest) and 3-36 (sag).

Where's the high or low point?

x* = -G_1·L / (G_2 - G_1) measured from BVC. For a sag, this is the low point (drainage low spot). For a crest, the high point. If x* is outside [0, L], the curve has no extremum and grades change monotonically.

Vertical Curve Calculator

Vertical Curve — Crest & Sag

Equal-tangent parabolic vertical curve. Length, K-factor, mid-curve offset, station and elevation of high or low point. AASHTO minimum K for stopping sight distance comparison.

Units

Curve type

Entering grade, G₁% (signed; positive = uphill)

Exiting grade, G₂%

Curve length, Lft

Design speed, Vmph (for AASHTO comparison)

PVI stationft (PVI = point of vert. intersection)

PVI elevationft

Algebraic difference, A—% (= G_2 − G_1)

Rate of curvature, K—ft / % (= L / |A|)

AASHTO min K (SSD)—ft / %

Mid-curve offset, e—ft

High/low point station—ft from BVC

High/low elevation—ft

Defaults: 3% upgrade transitioning to 2% downgrade over 500 ft, design speed 55 mph. AASHTO K_min table values for SSD govern minimum length.

$$ A = G_2 - G_1 \text{ (%)}, \quad K = L / |A|, \quad e = \frac{A L}{800} $$

Equal-tangent parabola from BVC (start) at x = 0:

$$ y(x) = G_1 \cdot x + \frac{(G_2 - G_1)}{2L} x^2 $$

High/low point (where dy/dx = 0):

$$ x^* = -\frac{G_1 \, L}{G_2 - G_1} $$

K rate of vertical curvature, ft per percent grade change (smaller is sharper) · A algebraic difference of grades · L total curve length, BVC to EVC · e offset from PVI to mid-curve · x distance from BVC.

K-factor — the design parameter

K = L / |A| is the most useful design parameter for vertical curves. It's the horizontal distance required to change the grade by 1% — and it's directly tied to sight distance. AASHTO Green Book Table 3-34 gives K_min vs design speed for crest curves (sight distance limited by line of sight over the crest) and Table 3-36 for sag curves (sight distance limited by headlight beam).

Crest K_min for SSD (AASHTO Table 3-34, level grade)

30 mph: K = 19
40 mph: K = 44
50 mph: K = 84
55 mph: K = 114
60 mph: K = 151
65 mph: K = 193
70 mph: K = 247

Sag K_min is approximately 30–50% larger than crest at typical speeds. Add comfort criterion (rider centripetal acceleration on sag) — for high-speed roads (≥ 50 mph), comfort sometimes governs.

Mid-curve offset

The maximum offset from the PVI tangent intersection to the curve is e = AL/800 (US, with A in % and L in ft) or e = AL/200 in SI metric. This is the critical clearance for overpasses and signs above crest curves; e flips sign for sag curves (curve is below the PVI grade).

High/low point and drainage

For a sag curve, the low point catches all surface drainage in cross-section. Offset gutter inlets at the low point ± 25–50 ft to spread the catchment. For a crest curve, no drainage point — but a flat spot in the cross-slope appears at the high point on superelevation transitions, requiring careful drainage design.

Equal-tangent vs unequal-tangent

This calculator assumes equal-tangent (BVC and EVC equidistant from PVI). Unequal-tangent curves (different lengths each side of PVI) are used to fit grade transitions to existing structures or right-of-way constraints. The formula generalizes — see AASHTO §3.4.6.

Reference: AASHTO (2018). A Policy on Geometric Design of Highways and Streets, 7th ed., §3.4. Garber, N.J., Hoel, L.A. (2015). Traffic and Highway Engineering, 5th ed., Cengage, ch. 16.