8.1 Horizontal Circular Curve Elements

Simple Circular Curves and Stationing

Horizontal curves are part of the official FS Survey Computations and Computer Applications knowledge area, and they are a natural place for calculation traps. A simple circular curve joins two tangents with one constant radius. The curve is defined by the point of curvature, point of tangency, point of intersection, radius, and central angle. Before reaching for formulas, draw the tangents and label PC, PI, PT, R, and delta.

The central angle delta is the angle between the two radii to the PC and PT, and it equals the intersection angle between tangents for a simple circular curve. Tangent length T is R tan(delta/2). Arc length L is pi R delta / 180 when delta is in degrees. Long chord LC is 2 R sin(delta/2). External distance E is R sec(delta/2) - R. Middle ordinate M is R - R cos(delta/2). These formulas are often listed in reference material, but the exam skill is knowing which value the problem asks for.

Stationing is a frequent FS application. If the PI station is 25+40.00 and T is 312.60 ft, the PC station is 25+40.00 - 3+12.60 = 22+27.40. If the curve length is 614.20 ft, the PT station is 22+27.40 + 6+14.20 = 28+41.60. Do not add curve length to the PI station unless the problem specifically defines a different route relationship. The PI is not on the arc.

Degree of curve may be defined by arc or chord, depending on local convention or problem statement. The FS exam may avoid ambiguity, but if D is given, confirm whether it is based on a 100 ft arc or a 100 ft chord. The radius relationship changes slightly. If the problem gives radius directly, use radius directly and do not introduce degree of curve.

Curve direction also matters. A curve to the right or left affects offsets and deflection signs, but it does not change the magnitudes of T, L, LC, E, or M. For staking, deflection angles from the tangent are usually proportional to arc length for equal stations along a simple curve. Keep the sign convention separate from the geometry magnitude.

A reliable curve workflow is:

1. Sketch PC, PI, PT, center, radius lines, and tangents.
2. Convert delta to decimal degrees if needed.
3. Compute T first if stationing is required.
4. Compute PC from PI minus T.
5. Compute L, then PT from PC plus L.
6. Compute chord, external, or ordinate only if asked.
7. Check whether the answer should be feet, meters, station format, or an angle.

Many wrong answers are one formula away from correct. A choice near 2 R sin(delta/2) when the problem asks for arc length is a long chord distractor. A choice that adds T and L directly to PI station may confuse tangent and curve stationing. The sketch protects against both errors.