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Vertical curve solver

Q: What sign do I give the grades?

The sign the profile carries: positive uphill in the direction of increasing stationing, negative downhill. Enter them as they are labelled on the sheet. The sign is not a convention you can choose — it decides whether the curve is a crest or a sag and where, or whether, the turning point falls.

Q: Why is there no high or low point on my curve?

Because both grades run the same way. A curve from +4% to +1% is climbing at both ends, so the grade never passes through zero between them and the extremes are the BVC and the EVC. The turning-point formula returns a value outside the curve in that case, and the calculator says so rather than printing a meaningless station.

Q: Is the high point under the PVI?

Only when the two grades are equal and opposite. Otherwise it is pulled towards the flatter grade. On a +3% to −2% curve it lands three-fifths of the way from the BVC, not half way. Assuming it sits at the PVI is a common and consequential error when the number is being used to place a drainage inlet.

Q: What is K used for?

It is the index into sight-distance design tables. For a crest curve the available stopping sight distance is governed by K rather than by the length and grade change separately, so a standard specifies a minimum K for a design speed and the required length follows as K times the absolute grade change. It is also a quick comparison of how gentle two curves are.

Q: Does this handle unequal-tangent curves?

No. This solves the equal-tangent case, where half the length lies each side of the PVI, which is the overwhelming majority of vertical curves. An unequal-tangent curve is two parabolas joined at a common point on the line between the tangent points, and it is computed as two separate curves that share that point.

Q: Can I work in meters?

Yes. The grades are percentages and the arithmetic is unit-free, so entering the length and elevations in meters gives elevations in meters. The one thing to watch is the station interval: this tool writes stations at a 100-unit interval, so a metric alignment stationed at 1000 m per station needs its stations converted, or read as plain distances.

Grades and length in; the turning point, the K value and a full grade-elevation table out.

Everything is computed in your browser. Nothing you type is uploaded, stored, or logged.

Entering grade g₁ (%)Signed: −3.2 is a downgrade

Exiting grade g₂ (%)Signed, as labelled on the profile

Curve length LHorizontal length, BVC to EVC

PVI station25+00.00, or a plain distance

PVI elevationElevation of the grade intersection

The curve is equal-tangent: half its length lies each side of the PVI. Elevations are in the same unit as the length, and the station interval is taken as 100 units.

Curve type: Crest
Grade change A = g₂ − g₁: -5.00%
BVC station: 22+00.00
BVC elevation: 441.000
EVC station: 28+00.00
EVC elevation: 444.000
External offset |A|L/8: 3.750
K = L / |A|: 120.00
Rate of grade change r: -0.83% / station
High point station: 25+60.00
High point elevation: 446.400

The turning point sits 360.00 past the BVC, which is where the grade passes through zero. It is not under the PVI, and it only coincides with the mid-point of the curve when the two grades are equal and opposite.

Grade elevations at full stations, with the BVC, EVC and turning point carried as labelled rows
Station	x from BVC	Tangent elevation	Offset	Curve elevation	Note
22+00.00	0.00	441.000	0.000	441.000	BVC
23+00.00	100.00	444.000	-0.417	443.583
24+00.00	200.00	447.000	-1.667	445.333
25+00.00	300.00	450.000	-3.750	446.250
25+60.00	360.00	451.800	-5.400	446.400	High point
26+00.00	400.00	453.000	-6.667	446.333
27+00.00	500.00	456.000	-10.417	445.583
28+00.00	600.00	459.000	-15.000	444.000	EVC

The tangent column is the back grade line extended straight through, and the offset column is the parabola pulling away from it. The offset at the PVI station is the external offset, and on a crest both are negative because the curve lies below the grade line it corrects.

A vertical curve joins two grades on a profile. It is defined by the two grades, the horizontal length of the curve, and the station and elevation of the point where the grades intersect. From those five numbers everything else follows: where the curve begins and ends, the elevation at any station along it, how far the curve departs from the grade lines, and where the profile crests or bottoms out.

Why a parabola and not a circle

A horizontal curve is a circular arc because a vehicle travelling it at constant speed needs constant lateral acceleration, and a circle is the curve of constant radius. A vertical curve is a parabola for the corresponding reason with a different answer: what matters going over a crest or through a sag is that the grade changes at a steady rate, so the vertical acceleration a driver feels is constant. The curve whose slope changes linearly with horizontal distance is the parabola, not the circle.

Three practical consequences follow. The rate of grade change, r, is a single constant for the whole curve, which is what a design standard can specify. Elevations are computed from a quadratic in x with no trigonometry at all, which is what made the method workable by hand for a century. And offsets from the tangent are proportional to the square of the distance from the tangent point, which gives the layout shortcut that older field books use throughout.

The curve equation

Measure x horizontally from the BVC — the beginning of the vertical curve, where the back grade leaves the tangent. Express the grades as decimals rather than percent. Then the elevation on the curve is:

elev(x) = elev_BVC + g₁·x + (g₂ − g₁)·x² / (2L)

The first two terms are the back tangent extended. The third is the offset from it.

That split is how the calculation is organised on paper and how the table on this page is laid out. The tangent elevation is the back grade run straight through, ignoring the curve; the offset is the correction that bends it. The offset is zero at the BVC, grows as the square of x, and at the EVC has grown to exactly the amount needed to land on the forward grade.

The curve is equal-tangent, meaning half its length sits each side of the PVI. So the BVC is L/2 back from the PVI along the entering grade and the EVC is L/2 ahead of it along the exiting grade, which is where the BVC and EVC elevations printed by the calculator come from. Unequal-tangent curves exist for constrained sites and are computed as two joined parabolas; this tool does not solve them.

A, r and K

Three derived quantities carry most of the design conversation about a vertical curve.

A = g₂ − g₁              algebraic grade change, percent
r = A / L               rate of grade change per unit length
K = L / |A|             length per percent of grade change

A is signed: negative is a crest, positive is a sag.

A is the total change in grade, taken algebraically. Going from +3% to −2% is a change of −5%, not of 1%, and the sign is what distinguishes a crest from a sag. A crest has A negative; a sag has A positive.
r is how fast the grade changes. Quoted per station it is the number a plan sheet prints, and it is what limits how abruptly a long vehicle is asked to pitch.
K is the length of curve required per percent of grade change. Sight-distance tables are indexed by K, because for a crest the available stopping sight distance depends on K and not on A and L separately. Given a required K from the design standard, the minimum curve length is K·|A|.

The high or low point

The turning point is where the grade on the curve reaches zero. Differentiating the curve equation and setting the result to zero gives its distance from the BVC directly.

x_turning = −g₁ · L / (g₂ − g₁)

Measured from the BVC. Valid only when 0 < x < L.

When the grades do have opposite signs the turning point is genuine, and it is not under the PVI. It falls at the PVI station only in the special case where the two grades are equal and opposite. On the default curve here, +3% into −2%, the high point sits 360 ft past the BVC on a 600 ft curve — three-fifths of the way along, pulled towards the flatter of the two grades.

The external offset

The vertical distance from the PVI to the curve is the single most useful number for checking clearance under a structure or over one.

E = |A| · L / 8

With A as a decimal grade change. It is the offset at x = L/2, where the quadratic term equals AL/8.

It also gives the classic layout shortcut. Because offsets go as the square of the distance from the tangent point, the offset at the quarter point of the curve is one quarter of the offset at the mid-point, and the offsets from either end are symmetric. A field book can therefore lay out a whole curve from E alone and a table of squares.

How a plan sheet lays this out

A profile sheet carries a grade table under the drawn profile, and the table on this page is built to the same shape. It runs at full stations across the curve, with three extra rows forced in regardless of where they fall: the BVC, the EVC, and the turning point when there is one. Those three are the rows a construction crew actually needs, and the odd stations they land on are exactly why they have to be carried explicitly rather than left to the interval.

Station — the chainage, in plus notation.
x from BVC — what the quadratic term is evaluated at. Zero at the BVC, L at the EVC.
Tangent elevation — the back grade run straight through, before any bending.
Offset — the correction: negative on a crest, positive in a sag, growing with x².
Curve elevation — the sum of the previous two, and the number a grade stake is set to.

Two arithmetic checks come free from that layout. The offset at the EVC must equal g₂·L − g₁·L over two, which is to say the curve must land exactly on the forward grade. And the curve elevation at the EVC computed here must equal the PVI elevation plus g₂·L/2 computed independently from the forward tangent. If they disagree, the length or one of the grades was entered wrong, and no amount of interpolating in the table will reveal it.

Questions

What sign do I give the grades?

The sign the profile carries: positive uphill in the direction of increasing stationing, negative downhill. Enter them as they are labelled on the sheet. The sign is not a convention you can choose — it decides whether the curve is a crest or a sag and where, or whether, the turning point falls.

Why is there no high or low point on my curve?

Because both grades run the same way. A curve from +4% to +1% is climbing at both ends, so the grade never passes through zero between them and the extremes are the BVC and the EVC. The turning-point formula returns a value outside the curve in that case, and the calculator says so rather than printing a meaningless station.

Is the high point under the PVI?

Only when the two grades are equal and opposite. Otherwise it is pulled towards the flatter grade. On a +3% to −2% curve it lands three-fifths of the way from the BVC, not half way. Assuming it sits at the PVI is a common and consequential error when the number is being used to place a drainage inlet.

What is K used for?

It is the index into sight-distance design tables. For a crest curve the available stopping sight distance is governed by K rather than by the length and grade change separately, so a standard specifies a minimum K for a design speed and the required length follows as K times the absolute grade change. It is also a quick comparison of how gentle two curves are.

Does this handle unequal-tangent curves?

No. This solves the equal-tangent case, where half the length lies each side of the PVI, which is the overwhelming majority of vertical curves. An unequal-tangent curve is two parabolas joined at a common point on the line between the tangent points, and it is computed as two separate curves that share that point.

Can I work in meters?

Yes. The grades are percentages and the arithmetic is unit-free, so entering the length and elevations in meters gives elevations in meters. The one thing to watch is the station interval: this tool writes stations at a 100-unit interval, so a metric alignment stationed at 1000 m per station needs its stations converted, or read as plain distances.

Sources

Washington State DOT, Highway Surveying Manual M 22-97, Chapter 11 — published state manual; the equal-tangent parabolic curve equation, the external offset and the grade-table layout
Montana DOT, Survey Manual, Appendix C — published state manual; vertical curve elements and the turning-point relationship

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