8.1 Horizontal Circular Curve Elements

The Simple Circular Curve and Its Six Elements

Horizontal curves sit in the FS Survey Computations and Computer Applications knowledge area, and they reward a candidate who sketches the geometry before reaching for a formula. A simple circular curve is an arc of constant radius R that connects two straight tangent lines. The tangents meet at the point of intersection (PI), the curve begins at the point of curvature (PC), and it ends at the point of tangency (PT). The angle between the two tangents — equal to the central angle Δ (delta) subtended by the arc — is the deflection of the alignment.

The six elements every FS candidate must produce on demand all flow from R and Δ:

Tangent length T is measured from the PI back to the PC, and by symmetry the PI-to-PT distance is the same T. The long chord LC is the straight line from PC to PT. The external distance E runs from the PI to the midpoint of the curve, and the middle ordinate M runs from that curve midpoint to the midpoint of the long chord. Sketching these once removes most sign and substitution errors before any arithmetic begins.

Degree of Curve: Arc vs. Chord Definition

The degree of curve D describes sharpness. Under the arc definition — standard for highways — D is the central angle that subtends a 100-ft arc, giving D = 5729.58/R (because a 100-ft portion of the full 2πR circumference is the fraction D/360). Under the chord definition — traditional in railroad work — D is the central angle that subtends a 100-ft chord, so sin(D/2) = 50/R. For a given R the chord-definition D is very slightly larger, and the two definitions diverge as curves get sharp. Always read which definition a problem states; mixing them is a classic FS trap.

A worked element computation

Given R = 1000 ft and Δ = 24°00'00":

• T = 1000·tan(12°) = 1000(0.21256) = 212.56 ft
• L = π(1000)(24)/180 = 418.88 ft
• LC = 2(1000)·sin(12°) = 2000(0.20791) = 415.82 ft
• E = 1000(1/cos12° − 1) = 1000(1.02234 − 1) = 22.34 ft
• M = 1000(1 − cos12°) = 1000(1 − 0.97815) = 21.85 ft
• D (arc) = 5729.58/1000 = 5°43.7'

Notice L > LC (the arc is longer than its chord) and E > M (the external pulls farther from the PI than the middle ordinate). These inequalities are quick sanity checks: any answer violating them is wrong.

Stationing the PC and PT

Route alignments carry continuous stationing along the tangents and the curve. The single most common stationing mistake is adding 2T instead of L. The correct procedure:

1. PC station = PI station − T. Stationing follows the tangent into the PC, so you subtract one tangent length.
2. PT station = PC station + L. Because the alignment now runs along the arc, you advance by the curve length L, not by the chord and not by 2T.

For R = 1000 ft, Δ = 24°, with PI at station 50+00.00: T = 212.56, so PC = 50+00.00 − 2+12.56 = 47+87.44; then PT = 47+87.44 + 4+18.88 = 52+06.32. If you had wrongly used 2T (425.12 ft) the PT would land at 52+12.56 — about 6 ft off, exactly the kind of small error the exam plants.

Angle-unit discipline

FS items often state Δ in degrees-minutes-seconds (DMS), e.g. 24°17'30". Convert to decimal degrees (24 + 17/60 + 30/3600 = 24.2917°) before any trig, and confirm your calculator is in degree mode, not radians. Memorize 1° = 60', 1' = 60". Keeping R and all distances in the same length unit (all feet or all meters) is equally important, because R appears as a direct multiplier in five of the six element formulas, so a single unit slip scales every result.

Compound, Reverse, and Spiral Curves

Beyond the simple curve, the FS exam expects familiarity with three related horizontal alignments. A compound curve is two or more circular arcs of different radii curving in the same direction, joined at a point of compound curvature (PCC). Each arc keeps its own R and Δ and is computed separately; the shared tangent point makes the arcs tangent to one another. A reverse curve is two arcs curving in opposite directions, meeting at a point of reverse curvature (PRC). Reverse curves are avoided on high-speed highways because the superelevation must reverse abruptly, but they appear in property and railroad work.

A spiral (transition) curve is an easement placed between a tangent and a circular arc so that curvature changes gradually rather than instantly. The standard form is the Euler spiral (clothoid), whose radius decreases in inverse proportion to the length traveled, giving a constant rate of change of curvature. Its purpose is to introduce superelevation (banking) and lateral acceleration smoothly, improving safety and rider comfort. Key spiral terms the exam may reference:

The spiral length Ls is selected by the designer based on design speed and the radius of the circular curve it joins; longer spirals suit higher speeds and sharper curves. On the FS exam, spiral questions are usually conceptual — recognizing the TS/SC/CS/ST sequence and the role of the spiral as a smooth easement — rather than full clothoid computation.