CELE Surveying (Geomatics) — Horizontal Curves (Simple, Compound, Reverse)Revision Notes

Formulas

Exam Tips

• Memorize the five formulas as a set: T(tan), Lc(pi), LC(sin), E(sec-1), M(1-cos). The pattern tan/sin/sec-1/1-cos at half-angle is easy to recall.
• In board exams, always box given values (R, I, Sta PI) and list unknowns before choosing a formula.
• E and M are related: E = M + M²/(2R) approximately, but use exact formulas in board exams.
• Check: E > M always (the PI is farther than the midchord point). If your E < M, you made an error.
• Quick sanity check for Lc: It must be greater than LC and less than πR (the full semicircle arc).

Key Points

• A simple circular curve has a single, constant radius R throughout its length.
• The angle I (also called the intersection angle or deflection angle) is the angle between the two tangent lines meeting at the PI. It equals the central angle subtended by the curve.
• The PC (Point of Curvature) is where the alignment leaves the first tangent and enters the curve.
• The PT (Point of Tangency) is where the curve ends and the second tangent begins.
• The PI (Point of Intersection) is where the two tangents, if extended, meet — the driver never actually passes through the PI on a curved road.
• The five curve elements (T, Lc, LC, E, M) are all functions of R and I only.
• The long chord LC is always less than Lc (arc length) for any curve with I > 0.
• For very flat curves (small I or large R), LC ≈ Lc, but for sharp curves the difference is significant.

Definitions

Section Title

1. Geometry of the Simple Circular Curve

Common Mistakes

• Using I instead of I/2 in sin, cos, tan functions — always halve the central angle before applying trigonometry.
• Forgetting to convert I to radians when using Lc = R × I_rad (or equivalently, using Lc = π R I / 180 to keep I in degrees).
• Confusing E (uses secant/reciprocal of cosine) with M (uses cosine directly). Remember: E is farther from the curve (at PI), so E > M always.
• Adding 2T to PI station to get PT — WRONG. Sta PT = Sta PC + Lc. The road goes around the arc, not along the tangents.
• Treating LC (long chord) and Lc (arc length) as the same — they are different. LC < Lc always.
• Ignoring units: if I is in degrees, use the π/180 factor in Lc; if I is already in radians, do not multiply again.

Formulas

Exam Tips

• The constant 1145.916 = 20 × 180 / π. You can re-derive it in seconds if you forget it during the exam.
• If D is given (e.g., '3-degree curve, arc definition'), immediately compute R = 1145.916/3 = 381.97 m, then proceed with standard curve element formulas.
• For compound or reverse curves, each curve has its own D (and thus its own R). Treat them separately.

Key Points

• The degree of curve D is the central angle (in degrees) subtended by a standard length of arc or chord.
• Philippine and metric surveying practice uses a 20-m standard length (either arc or chord).
• US practice uses a 100-ft (≈ 30.48 m) chord or arc; do not mix systems in board problems.
• Arc Definition (most common in Philippine boards): D is the central angle for a 20-m arc. Gives R = 1145.916 / D.
• Chord Definition: D is the central angle for a 20-m chord. Gives sin(D/2) = 10/R, so R = 10 / sin(D/2).
• For flat curves (small D, large R), arc and chord definitions give nearly identical R values. For sharp curves (large D), the difference is notable.
• Knowing which definition is being used is critical — the problem will specify or it can be inferred from context.

Definitions

Section Title

2. Degree of Curve — Arc and Chord Definitions

Common Mistakes

• Dividing 1145.916 by the wrong quantity — the formula is R = 1145.916/D, not D = 1145.916/R (though that rearrangement is also valid).
• Using the chord definition formula when the problem implies arc definition (or vice versa) — read the problem carefully.
• Forgetting that 1145.916 is derived from a 20-m station length. In older US problems with 100-ft stations, the constant is 5729.58 ft/degree (or 1745.33 m/degree) — do not mix.

Formulas

Exam Tips

• The golden rule: Sta PC = PI − T; Sta PT = PC + Lc. Write these two lines first in any stationing problem.
• If the answer choices cluster near a value obtained by adding 2T to PI, recognize this as the distractor for the classic 'add arc, not tangents' error.
• Always verify: Sta PT > Sta PC (curve runs forward); Sta PT > Sta PI if T < Lc/2 (common for large I).

Key Points

• Stations are continuous chainage distances (in meters) measured along the road centerline from a reference origin (usually Sta 0+000).
• Notation: Sta 10+120 means 10,120 m from the origin. The '+' separates kilometers from meters.
• The PI station is usually given. From it, subtract T to get the PC station.
• Add the arc length Lc (NOT 2T, NOT the long chord LC) to the PC station to get the PT station.
• Stationing follows the actual path of travel — along the curve, not across chords or back along tangents.
• If multiple curves are involved (compound or reverse), stationing is continuous: the PT of curve 1 is the PCC (Point of Compound Curvature) or PRC (Point of Reverse Curvature) and starting station for curve 2.

Definitions

Section Title

3. Stationing Along the Alignment

Common Mistakes

• Using Sta PT = Sta PI + T (WRONG — this gives a point on the forward tangent, not the PT of the curve).
• Using Sta PT = Sta PI − T + 2T = Sta PI + T (same wrong result approached differently).
• Using LC (chord length) instead of Lc (arc length) to add to PC — always use the arc.
• Writing stations incorrectly: 10+010.81 ≠ 10+010.8 when precision matters in computation checks.

Formulas

Exam Tips

• In compound curve problems, always identify and list: R1, R2, I1, I2 separately before computing. Label a diagram.
• Sta PT = Sta PC + Lc1 + Lc2 — simply add both arc lengths to the PC station.
• If only the total I and one R are given, you likely need additional constraints (e.g., equal tangents or a given PC-PT distance) to solve — look for these in the problem stem.

Key Points

• A compound curve consists of two (or more) simple circular arcs of different radii (R1, R2) turning in the same direction, joined at a common tangent point called the PCC.
• The PCC is simultaneously the PT of the first curve and the PC of the second curve.
• The overall deflection angle I = I1 + I2, where I1 and I2 are the central angles of the first and second curves respectively.
• Each curve has its own T, Lc, LC, E, M computed independently using its own R and I.
• The common (or 'middle') tangent at the PCC is tangent to both curves simultaneously.
• Compound curves are used where terrain, right-of-way, or alignment controls prevent a single radius — common in mountain roads like those in the Benguet or Mountain Province highways in the Philippines.
• The tangent lengths from PI to PC and from PI to PT are not equal unless R1 = R2 (simple curve case).

Definitions

Section Title

4. Compound Curves

Common Mistakes

• Treating the compound curve as a single curve and using a combined I with a single R — each curve must be handled separately.
• Stationing error: Sta PT = Sta PC + Lc1 + Lc2 (correct), not Sta PC + T1 + T2.
• Confusing the sum I = I1 + I2 with a single curve's I when computing elements.

Formulas

Exam Tips

• When the problem says 'parallel tangents' + 'reverse curve', immediately write: I1 = I2 = I and use d = (R1+R2)(1−cosI) [or 2R(1−cosI) if equal radii].
• Draw the reverse curve S-shape on scratch paper: label PC1, PRC, PT2, the two centers O1 and O2 on opposite sides, and the offset d. This prevents sign and direction errors.
• For the chord from PC1 to PT2 with equal radii and parallel tangents: L = 2R sinI (derived from the isoceles triangle formed by O1, PC1, and PRC).

Key Points

• A reverse curve consists of two circular arcs turning in opposite directions (one left, one right), joined at a common point called the PRC (Point of Reverse Curvature).
• The overall alignment forms an S-shape in plan view.
• At the PRC, the two curves share a common tangent but curve away from each other — creating an abrupt change in direction of centrifugal force.
• On high-speed roads, a tangent section is inserted between the two reverse curves to allow for superelevation runoff and driver reaction — a 'tangent-separated reverse curve.'
• For parallel tangents (a common board exam scenario), a geometric relationship between R1, R2, and the perpendicular offset d between tangents is used: d = Lc1×sinI1 / (2) type relationships derived from geometry.
• The most common special case: parallel tangents with equal radii (R1 = R2 = R), giving d = 2R(1−cosI) where I is the central angle of each curve.
• Reverse curves are prohibited on high-speed highways (design speed > 80 km/h per DPWH Design Guidelines) without a tangent between them.
• Stationing: Sta PRC = Sta PC1 + Lc1; Sta PT = Sta PRC + Lc2.

Definitions

Section Title

5. Reverse Curves

Common Mistakes

• Forgetting that parallel tangents force I1 = I2 — students sometimes try to solve with different central angles and get an unsolvable system.
• Confusing d (perpendicular distance between parallel tangents) with the chord length or arc length.
• Using the same formula for compound and reverse curves — these are fundamentally different geometric configurations.
• Applying reverse curve formulas to equal-radii cases with the general (R1 ≠ R2) formula instead of the simplified one — check if R1 = R2 first.