10.3 Compound and Reverse Curves
Key Takeaways
- A compound curve is two same-hand arcs meeting at a PCC with centers |R1 − R2| apart; a reverse curve is opposite-hand, meeting at a PRC with centers R1 + R2 apart.
- For R1 = 600 ft, Δ1 = 24°, R2 = 400 ft, and Δ2 = 18°, L1 = 251.33 ft, L2 = 125.66 ft, Ta = 215.69 ft, and Tb = 179.39 ft — not the simple-curve value 600 tan 21° = 230.32 ft.
- Equal-radius reverse curves R = 200 ft connecting parallel centerlines 80.00 ft apart have Δ = 36°52'12" on each arc and L = 128.70 ft each.
- Compound curves are the everyday ramp tool; reverse S-curves appear on offset streets; high-speed reverse usually needs a tangent between the arcs for superelevation runoff.
- The standing CES trap is applying T = R tan(I/2) to a compound as if it were one simple curve.
A simple curve has one radius. Domain III.E of the CES test plan also names compound and reverse curves. Both are two circular arcs sharing a common tangent at the junction, but they differ in hand (direction of curvature) and in the distance between centers. The exam trap is to grab T = R tan(Δ/2) as if the whole alignment were one simple curve.
Same-hand compound vs opposite-hand reverse
Compound curves (same direction)
A compound curve joins two (or more) circular arcs that turn the same way, with different radii, meeting at the point of compound curvature (PCC). At the PCC the two arcs share a common tangent and a common radial line. Centers O1, O2, and the PCC are collinear, and
O1O2 = |R1 − R2|.
Total intersection angle of the combined alignment is I = Δ1 + Δ2.
Each arc still has its own simple-curve length:
L1 = R1 Δ1 π / 180, L2 = R2 Δ2 π / 180, L = L1 + L2.
What is not simple is the overall tangent from the PC to the combined PI, or from that PI to the PT. Those incoming and outgoing tangents Ta and Tb are unequal when R1 ≠ R2. They are not T = R1 tan(I/2), not T = R2 tan(I/2), and not T = R_avg tan(I/2). Each local arc would have its own mini-PI if you treated it as a standalone simple curve (t1 = R1 tan(Δ1/2), t2 = R2 tan(Δ2/2)), and those local tangent lengths are also not Ta and Tb.
Place the PC at the origin with the incoming tangent along +x. Then, with I = Δ1 + Δ2,
PTy = R1 − (R1 − R2) cos Δ1 − R2 cos I
PTx = (R1 − R2) sin Δ1 + R2 sin I
Ta = PTx − PTy / tan I
Tb = PTy / sin I
Those two-center formulas are the replacement for T = R tan(I/2). Stationing still uses PC = PI − Ta and PT = PC + L1 + L2, never PI + Tb as a station.
Worked compound layout: R1 = 600 ft, Δ1 = 24°00', R2 = 400 ft, Δ2 = 18°00'
Place the larger radius first (common on ramps that tighten). I = 42°00'.
- L1 = 600 × 24 × π / 180 = 251.33 ft
- L2 = 400 × 18 × π / 180 = 125.66 ft
- L = 376.99 ft
Local simple tangents (not the overall PI tangents):
- t1 = 600 tan 12° = 127.53 ft
- t2 = 400 tan 9° = 63.35 ft
Two-center geometry for the combined PI:
- Ta = 215.69 ft (PC to PI along the back tangent)
- Tb = 179.39 ft (PI to PT along the forward tangent)
The trap value T = R1 tan(I/2) = 600 tan 21° = 230.32 ft is 14.63 ft too long on the incoming side. The average-radius fake T = 500 tan 21° = 191.93 ft is too short. Either error shifts every station.
Station the curve from a PI at 30+00.00:
- PC = 30+00.00 − 2+15.69 = 27+84.31
- PCC = 27+84.31 + 2+51.33 = 30+35.64
- PT = 30+35.64 + 1+25.66 = 31+61.30
Check: PC + L = 27+84.31 + 3+76.99 = 31+61.30. Note that PI + Tb = 30+00 + 1+79.39 = 31+79.39, which is not the PT station — the same PI+T trap, now with unequal Ta and Tb.
Arc degrees: D1 = 5729.58 / 600 = 9.549°, D2 = 5729.58 / 400 = 14.324°. The alignment gets sharper after the PCC, which is the usual ramp story: a large-radius entrance compounds into a smaller-radius turning roadway.
Reverse curves (opposite direction)
A reverse curve joins two circular arcs that turn opposite ways, meeting at the point of reverse curvature (PRC) or with a short tangent between. At a direct PRC the arcs share a common tangent, the hand flips, centers O1, O2, and the PRC are collinear, and
O1O2 = R1 + R2.
Net direction change of the alignment is |Δ1 − Δ2| when the two deflections partly cancel. Connecting parallel tangents requires Δ1 = Δ2 = Δ, and the offset between those tangents is
p = R1(1 − cos Δ) + R2(1 − cos Δ) = (R1 + R2)(1 − cos Δ).
Worked reverse layout: parallel streets, equal radii
Two parallel street centerlines are 80.00 ft apart. Connect them with a reverse pair of equal-radius arcs, R1 = R2 = 200.00 ft, meeting at a PRC (no intermediate tangent).
- p = 2 R (1 − cos Δ) = 80.00
- 1 − cos Δ = 80 / 400 = 0.20
- cos Δ = 0.80
- Δ = arccos(0.80) = 36.8699° = 36°52'12" (the 3-4-5 angle: sin Δ = 0.60, tan Δ = 0.75)
Each arc: L = 200 × 36.8699 × π / 180 = 128.70 ft Each simple tangent: T = 200 tan(18°26'06") = 200 × 1/3 = 66.67 ft Total centerline through both arcs: 257.40 ft Offset check: 400 (1 − 0.80) = 80.00 ft.
If the street design inserts a 120-ft internal tangent between PT1 and PC2 — the usual highway preference — you no longer have a single PRC. Each arc is then an ordinary simple curve with its own PC, PI, and PT, and stationing adds the 120-ft tangent between PT1 and PC2. Example: R1 = 400 ft, R2 = 350 ft, Δ = 20°00' each, connecting tangent 120.00 ft:
- T1 = 400 tan 10° = 70.53 ft, L1 = 139.63 ft
- T2 = 350 tan 10° = 61.71 ft, L2 = 122.17 ft
- Occupied centerline from PC1 to PT2 = L1 + 120.00 + L2 = 381.80 ft (the approach T1 and exit T2 sit outside that run)
Streets vs ramps
| Setting | Typical curve | Why |
|---|---|---|
| Local / collector streets | Reverse (S-curve) to pick up an offset, a dog-leg, or a parking-aisle shift | Low speed; parallel-street connections are common |
| Arterial that must tighten between controls | Compound | Fit a changing radius between right-of-way pins |
| Freeway ramps and loops | Compound, often large R then small R | Match high-speed entry to a tighter turning roadway |
| High-speed reverse | Avoid a bare PRC; insert a tangent | Needs length for superelevation to run down, through zero, and up the other way |
Caltrans-style highway practice generally does not like a reverse pair with no tangent between at highway speed. Direct PRC reverse curves still appear on slow streets, site drives, and some ramp gore geometry. Compound curves are the everyday ramp tool and appear on channelized intersection returns.
Exam traps
- Simple T on a compound. T = R tan(I/2) using R1, R2, or an average is wrong for Ta and Tb. Use two-center Ta, Tb, then station from PC through PCC to PT.
- Using O1O2 = |R1 − R2| on a reverse. Reverse centers are R1 + R2 apart.
- Forgetting the hand change. Compound: Δ1 + Δ2 add. Reverse: they oppose.
- Stationing through the combined PI. Same rule as Section 10.2: PC = PI − Ta, PT = PC + L1 + L2, not PI + Tb.
- Degree-of-curve mix. Each arc has its own D = 5729.58 / R. There is no single D for a compound.
If a CES stem gives only one R and one Δ, it is a simple curve. If it names a PCC or two radii in the same direction, it is compound. If it names a PRC, an S-curve, or opposite deflection, it is reverse. Draw O, the common radius through the junction, and the two tangents before you pick a formula.
A compound curve is best described by which geometry?
Equal-radius reverse curves (R = 200 ft each) connect parallel centerlines 80.00 ft apart at a PRC. Each arc's Δ is nearest to which value?
A compound curve has R1 = 600 ft, Δ1 = 24°, R2 = 400 ft, Δ2 = 18° (I = 42°). Incoming tangent PC to the combined PI is 215.69 ft. Which value is the simple-curve trap you must not use as Ta?