Horizontal Circular Curve Geometry

Assumptions

• Simple circular curve: Δ is the deflection angle between the tangents, equal to the central angle subtended by the arc.
• Degree of curve uses the ARC definition (angle subtending 100 ft of arc, D = 5729.58/R with R in feet) — the standard for highway work; the railroad chord definition differs.
• Stationing is US survey stationing in feet ("12+50" = 1250 ft), increasing in the direction of travel.

Solution steps

1. Central angle

   Express Δ in radians for the arc-length formula.

   Δrad = Δ·π/180

   Δ = 30 deg = 0.523599 rad

2. Tangent length

   Distance from PC (or PT) to the PI along the tangent.

   T = R·tan(Δ/2)

   T = 500 ft × 0.267949 = 133.975 ft

   = 134 ft

3. Curve (arc) length

   Length along the arc from PC to PT — this, not the chord, carries the stationing.

   L = R·Δ with Δ in radians (equivalently L = 100·Δ/D in ft)

   L = 500 ft × 0.523599 = 261.799 ft

   = 261.8 ft

4. Long chord

   Straight-line distance from PC to PT.

   LC = 2·R·sin(Δ/2)

   LC = 2 × 500 ft × 0.258819 = 258.819 ft

   = 258.8 ft

5. External distance

   From the PI to the midpoint of the curve, along the bisector.

   E = R·(1/cos(Δ/2) − 1)

   E = 500 ft × (1/0.965926 − 1) = 17.6381 ft

   = 17.64 ft

6. Middle ordinate

   From the midpoint of the long chord to the midpoint of the curve.

   M = R·(1 − cos(Δ/2))

   M = 500 ft × (1 − 0.965926) = 17.0371 ft

   = 17.04 ft

7. Degree of curve

   Arc-definition sharpness: the central angle subtending 100 ft of arc.

   D = 5729.58/R(ft), degrees per 100 ft of arc

   D = 5729.58 / 500 ft = 11.459 degrees per station

   = 11.46

8. Curve stationing

   The PC sits a tangent length BEFORE the PI; the PT is then reached along the ARC. Classic trap: PT = PC + L, NOT PI + T — station distance accumulates along the curve, so adding T at the PI overshoots.

   PC = PI − T; PT = PC + L (along the arc, never PI + T)

   PI at 1250 ft; PC = 1250 ft − 133.975 ft = 1116.03 ft; PT = 1116.03 ft + 261.799 ft = 1377.82 ft

   = 1377.82 ft

Results

Assumptions

• Simple circular curve: Δ is the deflection angle between the tangents, equal to the central angle subtended by the arc.
• Degree of curve uses the ARC definition (angle subtending 100 ft of arc, D = 5729.58/R with R in feet) — the standard for highway work; the railroad chord definition differs.

Solution steps

1. Central angle

   Express Δ in radians for the arc-length formula.

   Δrad = Δ·π/180

   Δ = 40 deg = 0.698132 rad

2. Tangent length

   Distance from PC (or PT) to the PI along the tangent.

   T = R·tan(Δ/2)

   T = 300 m × 0.36397 = 109.191 m

   = 109.2 m

3. Curve (arc) length

   Length along the arc from PC to PT — this, not the chord, carries the stationing.

   L = R·Δ with Δ in radians (equivalently L = 100·Δ/D in ft)

   L = 300 m × 0.698132 = 209.44 m

   = 209.4 m

4. Long chord

   Straight-line distance from PC to PT.

   LC = 2·R·sin(Δ/2)

   LC = 2 × 300 m × 0.34202 = 205.212 m

   = 205.2 m

5. External distance

   From the PI to the midpoint of the curve, along the bisector.

   E = R·(1/cos(Δ/2) − 1)

   E = 300 m × (1/0.939693 − 1) = 19.2533 m

   = 19.25 m

6. Middle ordinate

   From the midpoint of the long chord to the midpoint of the curve.

   M = R·(1 − cos(Δ/2))

   M = 300 m × (1 − 0.939693) = 18.0922 m

   = 18.09 m

7. Degree of curve

   Arc-definition sharpness: the central angle subtending 100 ft of arc.

   D = 5729.58/R(ft), degrees per 100 ft of arc

   D = 5729.58 / 984.252 ft = 5.8213 degrees per station

   = 5.821

Results

Assumptions

• Simple circular curve: Δ is the deflection angle between the tangents, equal to the central angle subtended by the arc.
• Degree of curve uses the ARC definition (angle subtending 100 ft of arc, D = 5729.58/R with R in feet) — the standard for highway work; the railroad chord definition differs.

Solution steps

1. Central angle

   Convert Δ from degrees-minutes-seconds to decimal degrees, then to radians for the arc-length formula.

   Δ = d + m/60 + s/3600, then Δrad = Δ·π/180

   Δ = 55° 30′ 0″ = 55.5 deg = 0.968658 rad

2. Radius from degree of curve

   With the arc definition, D degrees of central angle subtend exactly 100 ft of arc, which fixes the radius.

   R = 18000/(π·D) = 5729.58/D, R in ft for D per 100 ft of arc

   R = 5729.58 / 4 = 1432.39 ft

   = 1432.39 ft

3. Tangent length

   Distance from PC (or PT) to the PI along the tangent.

   T = R·tan(Δ/2)

   T = 436.594 m × 0.526125 = 229.703 m

   = 229.7 m

4. Curve (arc) length

   Length along the arc from PC to PT — this, not the chord, carries the stationing.

   L = R·Δ with Δ in radians (equivalently L = 100·Δ/D in ft)

   L = 436.594 m × 0.968658 = 422.91 m

   = 422.9 m

5. Long chord

   Straight-line distance from PC to PT.

   LC = 2·R·sin(Δ/2)

   LC = 2 × 436.594 m × 0.465615 = 406.569 m

   = 406.6 m

6. External distance

   From the PI to the midpoint of the curve, along the bisector.

   E = R·(1/cos(Δ/2) − 1)

   E = 436.594 m × (1/0.884988 − 1) = 56.7394 m

   = 56.74 m

7. Middle ordinate

   From the midpoint of the long chord to the midpoint of the curve.

   M = R·(1 − cos(Δ/2))

   M = 436.594 m × (1 − 0.884988) = 50.2137 m

   = 50.21 m

Results