The hook
A horizontal curve is geometry: two tangent lines meeting at a PI, joined by a circular arc. Five parameters describe it (T, L, R, Δ, LC) and they all derive from each other. Sweep the sliders below and watch how T explodes as Δ approaches 180° — that's why sharp curves need long approach tangents.
T (tangent)86.60 ftR · tan(Δ/2)
L (arc)157.08 ftR · Δ (rad)
LC (chord)150.00 ft2R · sin(Δ/2)
M (mid ord)20.10 ftR · (1 - cos Δ/2)
E (external)23.21 ftR · (sec Δ/2 - 1)
Try this: sweep Δ from small to large with R fixed, then sweep R with Δ fixed. Notice how T grows much faster than L when Δ approaches 180° — that's why sharp curves need long approach tangents.
Memorize these
Concepts that show up on the exam
PI (Point of Intersection)
Where the back tangent and forward tangent would meet if extended. The deflection angle Δ is measured here.
PC / PT
Point of Curvature (where the curve begins) and Point of Tangency (where it ends). Both are exactly distance T from the PI.
Δ (Delta)
The deflection angle — the angle between back tangent and forward tangent. Defines how much the road turns.
R (Radius)
Radius of the circular curve. Larger R → flatter curve → faster design speed.
T (Tangent length)
Distance from PC (or PT) to the PI. T = R · tan(Δ/2). Goes to infinity as Δ → 180°.
L (Curve length)
Arc length from PC to PT. L = R · Δ (Δ in radians). Always less than 2T for any real curve.
LC (Long chord)
Straight-line distance from PC to PT. LC = 2R · sin(Δ/2).
M (Middle ordinate)
Distance from the midpoint of LC to the midpoint of the arc. M = R · (1 − cos(Δ/2)).
E (External distance)
Distance from PI to the midpoint of the arc. E = R · (sec(Δ/2) − 1).
Keep these in muscle memory
Formulas to know cold
Tangent length
T = R · tan(Δ/2)Curve length (arc)
L = R · Δ (Δ in radians)Long chord
LC = 2R · sin(Δ/2)Middle ordinate
M = R · (1 − cos(Δ/2))External
E = R · (sec(Δ/2) − 1)Stationing (PT)
sta_PT = sta_PI − T + LPT comes back UP-station from PI by (L − T), not down by T.
Try it before you peek
Worked example
The problem
A horizontal curve has Δ = 32° and R = 800 ft. The PI is at station 12+50.00. Compute T, L, the station of the PC, and the station of the PT.
Don't fall for these
What trips people up
PT station comes from PC, not PI
The PT is L feet ALONG THE ARC from the PC. Adding L to the PI station overshoots by (L − T) every time.
Δ in degrees vs. radians
Tangent and chord formulas use Δ/2 inside trig functions (any unit your calculator is in). The arc length formula uses Δ in radians. Mix this up and L is off by ~57×.
Degree of curve confusion
"Degree of curve" is the central angle subtended by either a 100-ft arc (arc definition, highway) or a 100-ft chord (chord definition, railroad). Two different relationships to R — always confirm which definition the plan set or problem is using.
Test yourself
How well did it stick?
A quick 5-question check on Horizontal Curves. See where you stand and what to review.