A valley curve is the one place on a road where the design problem only appears after sunset.
By day, a dip in the road hides nothing — you can see straight across it to the far side. At night, everything changes. Your headlights point slightly downward, and as the road curves upward away from you, the beam strikes the surface much sooner than you would like. The lit distance ahead shrinks dramatically.
That is why valley curves are designed around headlight sight distance, and this post works through the whole thing.
What a Valley Curve Is
A valley curve, also called a sag curve, is a vertical curve whose convexity points downwards — a dip rather than a hump.
The Four Ways It Forms
A valley curve appears wherever two gradients meet in any of these four arrangements:
- A descending gradient meets another descending gradient (the road keeps falling, but less steeply)
- A descending gradient meets a flat gradient (the fall levels out)
- A descending gradient meets an ascending gradient (the classic dip)
- An ascending gradient meets another ascending gradient (the climb gets steeper)
There is one pattern behind all four. In every case the grade becomes algebraically greater as you travel along — from −5 % to −2 %, or from −3 % to +2 %, or from +2 % to +5 %. That increase is what makes the shape concave.
The Shape Used to Set It Out
A valley curve is made fully transitional, using two matching transition curves of equal length — one easing into the dip and one easing out of it. Each is set out as a cubic parabola:
y = b x3 where b = 2N / (3 L2)
In practice, though, road gradients are gentle. Where the deviation angle is small, a simple parabola is used for the design instead, and the results are perfectly satisfactory.
Why Headlight Sight Distance Governs
The rule is simple: the curve must be long enough — that is, gentle enough — that the headlight beam reaches at least as far as the stopping sight distance.
If the dip is too abrupt, the road rises into the beam quickly and the driver cannot see far enough ahead to stop.
Two geometric situations arise, and each needs its own formula:
- The curve is longer than the stopping sight distance (L > S)
- The curve is shorter than the stopping sight distance (L < S)
Case A: Curve Longer Than SSD (L > S)
When the curve is long, the whole of the driver’s sight distance falls within it.
The worst position is the very bottom of the dip. Why? Because of how a transition curve is built. At its start the radius is infinite — the road is effectively straight there. At the bottom the radius is at its minimum, meaning the road is curving most sharply upward. The sharper the upward curve, the sooner it intercepts the headlight beam, and the shorter the visible distance.
Working it out
For a small deviation angle, the curve is described by:
y = (N / 2L) x2
The beam leaves the headlight at height h1 and rises at angle α, so at a distance S ahead it sits at a height of (h1 + S tan α). Setting that equal to the height of the road at the same point:
h1 + S tan α = N S2 / (2L)
L = N S2 / [ 2 (h1 + S tan α) ]
Putting in the IRC values
IRC takes the headlight height as h1 = 0.75 m and the beam’s upward inclination as α = 1 degree:
2(h1 + S tan α) = 2(0.75) + 2S tan 1° = 1.5 + 0.035 S
L = N S2 / (1.5 + 0.035 S) when L > S
with S the SSD in metres and N the deviation angle in radians.
Note on SSD: ignore the gradient effect when computing S here. The critical visibility occurs at the lowest point of the dip, where the road is momentarily level.
Case B: Curve Shorter Than SSD (L < S)
Now the curve is short and the driver’s sight distance extends beyond it.
Here the worst position is the start of the curve, and the reasoning is worth following:
- With the vehicle at the bottom of the dip, the beam shoots out well past the end of the curve — visibility is generous.
- With the vehicle at the beginning of the curve, the beam only just clears the curve — visibility is at its worst.
So the formula is derived with the vehicle placed at the start.
Working it out
h1 + S tan α = (S − L/2) N
L = 2S − (2h1 + 2S tan α) / N
Substituting h1 = 0.75 m and α = 1° gives the same bracket as before:
L = 2S − (1.5 + 0.035 S) / N when L < S
This expression is approximate, but since real gradients are very small it is accurate enough for all practical purposes.
Which Case Applies? Do Both
Here is the awkward part. Each formula was derived assuming a particular relationship between L and S — but L is the very thing you are trying to find. You cannot know in advance which one applies.
The method: calculate both, then keep the one that satisfies its own assumption.
- Compute L from the Case A formula. If the answer really does come out greater than S, the assumption held and this is your answer.
- Compute L from the Case B formula. If the answer really is less than S, the assumption held and this is your answer.
- Only one of the two will be self-consistent. That is the length to adopt.
This self-checking approach is not a shortcut or a guess — it is the correct method, and marks are usually awarded for showing both calculations.
Notation
| Symbol | Meaning | Value or Unit |
|---|---|---|
| L | Length of the valley curve | m |
| S | Stopping sight distance | m |
| N | Deviation angle | radians |
| h1 | Height of the headlight beam | 0.75 m |
| α | Upward inclination of the beam | 1 degree |
| b | Cubic parabola constant | 2N/(3L2) |
Formula Summary
| Quantity | Expression |
|---|---|
| Cubic parabola | y = bx3, b = 2N/(3L2) |
| Parabolic ordinate | y = (N/2L)x2 |
| Case A general (L > S) | L = NS2/[2(h1 + S tan α)] |
| Case A with IRC values | L = NS2/(1.5 + 0.035S) |
| Case B general (L < S) | L = 2S − (2h1 + 2S tan α)/N |
| Case B with IRC values | L = 2S − (1.5 + 0.035S)/N |
Quick Revision Notes
- A valley or sag curve has convexity downwards.
- Four ways it forms; in each, the grade becomes algebraically greater along the road.
- Made fully transitional using two matching transition curves of equal length.
- Set out by a cubic parabola y = bx3 with b = 2N/(3L2); a simple parabola is used for small deviation angles.
- The governing criterion is headlight sight distance, since the problem occurs at night.
- h1 = 0.75 m and α = 1°, which together produce the bracket 1.5 + 0.035S.
- Case A (L > S): L = NS2/(1.5 + 0.035S). Worst position is the bottom of the dip.
- Case B (L < S): L = 2S − (1.5 + 0.035S)/N. Worst position is the start of the curve.
- Compute both cases and adopt the self-consistent one.
- Neglect the grade effect when computing SSD for a valley curve.
Mistakes Students Commonly Make
- Putting N in as a percentage. A deviation of 5 % goes into the formula as 0.05, not 5. This single error wrecks more answers than anything else here.
- Swapping the two formulas. Case A has S2 on top; Case B begins with 2S.
- Guessing which case applies instead of computing both.
- Using the 1.2 m driver eye height. On a valley curve the relevant height is the headlight at 0.75 m.
- Bringing in the 0.15 m object height. That belongs to summit curves, not to headlight sight distance.
- Applying a gradient correction to the SSD. It should be left out for valley curves.
- Doubling the 0.035 term. It is already 2 tan 1°, so the doubling is built in.
Conclusion
The valley curve is designed for the moment when a driver’s world shrinks to the length of a headlight beam. The curve must be gentle enough that the beam still reaches the stopping sight distance ahead. Two geometries are possible — curve longer than SSD, curve shorter than SSD — and both are built on the same pair of IRC constants, a headlight 0.75 m above the road tilted 1 degree upward. Compute both, keep the consistent one, keep N in radians, and this topic turns into reliable marks.
Frequently Asked Questions
What is a valley curve?
A vertical curve with convexity downwards, also called a sag curve, formed where two gradients meet in a concave arrangement.
In how many ways can a valley curve form?
Four: descending meeting descending, descending meeting flat, descending meeting ascending, and ascending meeting ascending.
Why is headlight sight distance the governing criterion?
Because a dip in the road hides nothing during daylight. The visibility problem only arises at night, when the driver can see no further than the headlight beam reaches.
What shape is used to set out a valley curve?
Two equal transition curves set out as a cubic parabola, y = bx3 with b = 2N/(3L2). For small deviation angles a simple parabola is used instead.
What headlight height and beam angle does IRC use?
A headlight height of 0.75 m and an upward beam inclination of 1 degree.
Where does the term 1.5 + 0.035S come from?
From 2(h1 + S tan α) with h1 = 0.75 m and α = 1 degree, giving 2(0.75) + 2S tan 1°.
What is the length formula when L is greater than S?
L = NS2/(1.5 + 0.035S), with S the SSD in metres and N the deviation angle in radians.
What is the formula when L is less than S?
L = 2S − (1.5 + 0.035S)/N.
How do you know which case to use?
You cannot know beforehand. Calculate both and adopt the one whose result satisfies its own condition — Case A must give L greater than S, Case B must give L less than S.
Should the gradient be allowed for in the SSD?
No. For a valley curve the grade effect is neglected, since the sight distance applies on both the falling and rising parts of the curve.