Horizontal Curves (Simple, Compound, Reverse) - Cheat Sheet
Your last-minute revision companion for horizontal curve geometry, stationing, and compound/reverse curve analysis. Focus on formula application, PC/PT calculation, and degree-of-curve conversions.
Simple Curve — Core Elements
Formulas
T = R tan(I/2)
Meaning
T = tangent distance (m); R = radius of curve (m); I = intersection/deflection angle (degrees)
Watch out
I MUST be in degrees here. Convert radians to degrees first (×180/π). Use tan(I/2), not tan(I).
When to use
Finding distance from PI back to PC or forward to PT; always your first calculation.
L_c = (π R I) / 180
Meaning
L_c = arc length of curve (m); π = 3.14159...; I = deflection angle (degrees)
Watch out
I MUST be in degrees. If I is in radians, use L_c = R × I (radians). Never use L_c = 2T in stationing.
When to use
Finding total length of the circular arc from PC to PT. Always used to find Sta PT.
LC = 2R sin(I/2)
Meaning
LC = long chord (straight-line distance from PC to PT); I = deflection angle (degrees)
Watch out
This is chord, not arc. LC < L_c always. Do not confuse with tangent distance T.
When to use
When chord distance (not arc) from start to end of curve is needed; also field layout.
E = R(sec(I/2) − 1)
Meaning
E = external distance (perpendicular offset from PI to the curve); I = deflection angle (degrees)
Watch out
sec(I/2) = 1/cos(I/2). Subtract 1 AFTER computing sec. E is always positive.
When to use
Finding distance from PI perpendicular to the curve; used in design/clearance checks.
M = R(1 − cos(I/2))
Meaning
M = middle ordinate (perpendicular distance from midpoint of chord to curve); I = deflection angle (degrees)
Watch out
M ≤ E always. Do not confuse with E (external). Use 1 − cos, not cos − 1.
When to use
Finding sagitta or mid-chord offset; used in curve verification and layout.
Sta PC = Sta PI − T
Meaning
Station of point of curvature = station of intersection minus tangent distance
Watch out
Subtract (not add) T from PI. PC is BEFORE the PI along the first tangent.
When to use
Always the first stationing step; PC is where curve begins.
Sta PT = Sta PC + L_c
Meaning
Station of point of tangency = station of PC plus arc length of curve
Watch out
Add L_c (arc), NOT 2T or T again. Sta PT = Sta PI + (T − L_c)? NO. Use Sta PT = Sta PC + L_c.
When to use
Second stationing step; PT is where curve ends and second tangent begins.
R = 1145.916 / D
Meaning
R = radius (m); D = degree of curve (degrees per 20 m arc); 1145.916 ≈ 180 × 20 / π
Watch out
This is ARC definition. If using chord definition, use sin(D/2) = 10/R instead. D and R are inversely proportional.
When to use
Converting degree of curve to radius, or vice versa (arc definition, 20 m standard).
D = 1145.916 / R
Meaning
D = degree of curve (degrees per 20 m arc); R = radius (m)
Watch out
Always state whether arc or chord definition. In Philippines, arc definition (20 m) is standard unless stated.
When to use
Converting radius to degree of curve; shaper curves have higher D.
Common values
Value
tan(I/2) varies; I = 30° → tan(15°) ≈ 0.268; I = 60° → tan(30°) ≈ 0.577
Symbol
I (degrees)
Quantity
Tangent distance factor
Value
1145.916 (for 20 m arc definition in SI units)
Symbol
k
Quantity
Arc conversion constant
Value
I = 20°, 30°, 40°, 45°, 60° (often multiples of 5° or 10°)
Symbol
I (degrees)
Quantity
Common deflection angles in design
Important facts
- The tangent distance T is measured along the tangent line, not perpendicular to PI.
- PC and PT are EQUIDISTANT from PI; Sta PT ≠ Sta PI + T (common mistake).
- Sta PT = Sta PC + L_c, NOT Sta PC + 2T.
- The arc length L_c is always GREATER than the chord LC (except when I = 0).
- External distance E ≤ T always; they are NOT the same.
- Middle ordinate M < E always.
- Deflection angle I and central angle are identical for a simple circular curve.
- The formula R = 1145.916/D uses the arc definition (20 m standard); chord definition yields a different constant.
- In the Philippines (NSCP references), the 20 m arc definition is standard unless explicitly stated as chord definition.
- For compound curves, the curves turn the SAME way; for reverse curves, they turn OPPOSITE ways.
Key definitions
PC (Point of Curvature)
Example
If PI is at 10+120 and T = 80 m, PC is at 10+040.
Start point of the circular curve; located T units back from PI along the first tangent.
PT (Point of Tangency)
Example
If PC is at 10+040 and L_c = 160 m, PT is at 10+200.
End point of the circular curve; located L_c units ahead of PC along the arc.
PI (Point of Intersection)
Example
If Sta PC = 10+040 and T = 80, then Sta PI = 10+120.
Intersection of the two tangent lines (before/after the curve); does NOT lie on the curve.
Deflection Angle (I)
Example
If two roads meet at 40°, the deflection angle I = 40°.
Interior angle between two tangent lines at PI; same as the central angle subtended by the arc.
Degree of Curve (D)
Example
D = 4° means a 20 m arc subtends 4° at the center; R = 1145.916/4 = 286.48 m.
Central angle (in degrees) subtended by a standard 20 m arc of the curve.
Radius of Curve (R)
Example
R = 300 m is a gentle curve; R = 80 m is sharp (e.g., parking lot exit).
Radius of the circular arc; larger R means gentler curve, smaller R means sharper curve.
Tangent Distance (T)
Example
If R = 300 m and I = 40°, T = 300 tan(20°) ≈ 109.19 m.
Distance along the tangent from PI to either PC or PT; symmetric.
Compound Curve
Example
A 400 m radius followed by a 250 m radius, both curving left.
Two circular curves of different radii in sequence, turning the SAME direction; share a common tangent point.
Reverse Curve
Example
A left-turn curve followed by a right-turn curve; used on winding mountain roads.
Two circular curves turning OPPOSITE directions (S-shape); typically separated by a short tangent for superelevation.
Diagrams to know
- Simple curve plan view showing PC, PI, PT, tangent lines, radius, and chord.
- Deflection angle I at PI with both tangent lines extended.
- External distance E perpendicular from PI to curve.
- Middle ordinate M from midpoint of chord to curve.
- Station markers at PC, PI, and PT with distances labeled.
- Degree of curve D showing 20 m arc and central angle.
Degree of Curve & Radius Conversion
Formulas
R = 1145.916 / D (arc definition, 20 m)
Meaning
R = radius (m); D = degree of curve per 20 m arc (degrees); constant = 180 × 20 / π
Watch out
Always specify arc or chord definition. This formula is for ARC definition only. Chord definition uses sin(D/2) = 10/R.
When to use
Converting D to R when degree of curve is given; standard in Philippines.
D = 1145.916 / R (arc definition, 20 m)
Meaning
D = degree of curve per 20 m arc (degrees); R = radius (m)
Watch out
Same caution: ARC definition only. Higher R → lower D (sharper curves have higher D).
When to use
Finding D when R is given.
sin(D/2) = 10 / R (chord definition, 20 m chord)
Meaning
D = degree of curve (central angle for 20 m chord); R = radius (m)
Watch out
Different from arc definition. Chord D slightly larger than arc D for same R. Verify which definition applies.
When to use
Chord definition conversion (less common in PH but may appear on exams).
Common values
Value
R = 1145.92 m
Symbol
D = 1°
Quantity
1° curve (arc def.)
Value
R = 572.96 m
Symbol
D = 2°
Quantity
2° curve
Value
R = 286.48 m
Symbol
D = 4°
Quantity
4° curve
Value
R = 143.24 m
Symbol
D = 8°
Quantity
8° curve
Important facts
- Degree of curve and radius are inversely proportional: D ∝ 1/R.
- Higher D = sharper curve; lower D = gentler curve.
- The arc and chord definitions yield slightly different D for the same R; arc D < chord D.
- In the Philippines, unless specified, assume ARC definition (20 m standard).
- A 1° curve is very gentle (R ≈ 1146 m); an 8° curve is quite sharp (R ≈ 143 m).
- D is always expressed as a positive number; no negative degrees.
- For horizontal highway design, typical D ranges from 0.5° to 12° (gentle to sharp).
Key definitions
Degree of Curve (Arc Definition)
Example
D = 3° → R = 1145.916/3 = 381.97 m.
Central angle (degrees) subtended by a 20 m arc; most common in Philippines.
Degree of Curve (Chord Definition)
Example
D = 3° (chord) → sin(1.5°) = 10/R → R ≈ 381.66 m (slightly different).
Central angle (degrees) subtended by a 20 m chord; used in some US/older specs.
Sharp Curve
Example
D = 8° → R = 1145.916/8 ≈ 143.24 m (sharp).
Curve with small radius (high D); sharper = smaller R and larger D.
Gentle Curve
Example
D = 1° → R = 1145.916/1 ≈ 1145.92 m (very gentle, near straight).
Curve with large radius (low D); gentler = larger R and smaller D.
Diagrams to know
- 20 m arc subtending central angle D at radius R.
- Relationship curve: D vs R (hyperbola; inverse proportionality).
- Comparison: arc definition vs chord definition for same D.
Stationing & Curve Layout
Formulas
Sta PC = Sta PI − T
Meaning
PC station = PI station minus tangent distance; PC is where curve begins.
Watch out
SUBTRACT T (not add). PC is BEFORE PI on the tangent line.
When to use
First step in all curve stationing; always use this.
Sta PT = Sta PC + L_c
Meaning
PT station = PC station plus arc length; PT is where curve ends.
Watch out
Add L_c (arc), NOT 2T or T. Do NOT use Sta PI + (T − L_c)—this is wrong.
When to use
Second step in stationing; PT marks end of circular curve.
Sta PT = Sta PI + T − L_c (alternative, less preferred)
Meaning
Alternative form: Sta PT can be computed from PI directly.
Watch out
This works mathematically but is error-prone; always use Sta PC + L_c instead.
When to use
If you forget the PC → PT formula; verify T > L_c is false for typical curves (T < L_c).
Station increment along arc = (L_c / 20) × D
Meaning
Each 20 m arc subtends D degrees; useful for setting out intermediate points.
Watch out
This is for equal-arc divisions. Equal-chord divisions use a slightly different approach.
When to use
Dividing the curve into equal-arc stations for field layout.
Important facts
- Stations are CUMULATIVE; they always increase along the route.
- PC marks the START of the curve; PT marks the END.
- From PI, PC is backward (subtract T); PT is forward (add L_c from PC).
- T is measured along the tangent, not perpendicular; T is usually > M and > E.
- For a typical 40° curve with R = 300 m: T ≈ 109 m, L_c ≈ 209 m; so PT is about 100 m PAST PI in terms of arc distance.
- Never skip the PC → PT calculation; always add L_c to Sta PC.
- When setting out a curve in the field, intermediate stations are spaced 20 m apart (or per project spec).
Key definitions
Stationing
Example
Sta 5+000 = 5000 m from origin; Sta 10+120 = 10,120 m.
Cumulative distance along a route from a reference point (origin); written as 10+240 means 10,240 m.
Station Format
Example
5+500 = 5.5 km; 12+085 = 12.085 km or 12,085 m.
Represented as 'km + m'; e.g., 10+240 means 10 km + 240 m = 10,240 m total.
Continuous Stationing
Example
If Sta PT = 10+220, the next tangent continues from 10+220 (not reset).
Stations increase monotonically along the route; they do not reset at curves.
Diagrams to know
- Linear stationing diagram with PI, PC, PT labeled and distances T, L_c marked.
- Station progression table: Sta PI → Sta PC → Sta PT with T and L_c shown.
- Curve layout showing intermediate stations at 20 m arc intervals.
Compound Curves
Formulas
No single formula; analyze as two simple curves in sequence
Meaning
Compound curve = Curve 1 (R₁, I₁) + Curve 2 (R₂, I₂), same direction, different radii.
Watch out
Do NOT use the single-curve formulas directly. Treat as two separate curves; compute PC₁, PT₁ = PCC, PT₂ separately.
When to use
When you encounter two curves turning the same way with different radii.
T_total = T₁ + T₂
Meaning
Total tangent distance ≈ sum of individual tangents (not exact, but approximation).
Watch out
This is approximate; exact calculation requires detailed geometry of the two curves and their common tangent point (PCC).
When to use
Rough estimate of total tangent offset for two-curve compound.
Important facts
- Compound curves turn the SAME direction (both left or both right).
- Radius changes at the PCC; deflection angles I₁ and I₂ are separate.
- Total deflection = I₁ + I₂.
- Use R₁ and I₁ to calculate Curve 1; use R₂ and I₂ for Curve 2.
- PCC = PT of Curve 1 = PC of Curve 2 (same station).
- Design practice: Compound curves used to transition from sharp to gentle curve (e.g., entering a highway).
- NSCP or road design standards specify minimum curve radii; compound curves help meet standards while fitting terrain.
Key definitions
Compound Curve
Example
A 400 m radius curve followed by a 250 m radius curve, both curving left; PCC is the transition point.
Two circular curves of different radii (R₁ ≠ R₂) in sequence, curving the SAME direction; share a common tangent point (PCC).
PCC (Point of Compound Curvature)
Example
If Curve 1 ends at Sta 8+500, then PCC = Sta 8+500 (also PT of Curve 1, also PC of Curve 2).
Common tangent point where the two curves meet; end of first curve and start of second.
First Tangent to Compound
Example
Road in a straight line until Sta PC₁.
Initial straight section before the first curve.
Final Tangent to Compound
Example
Road straightens out after Sta PT₂.
Final straight section after the second curve.
Diagrams to know
- Plan view of compound curve showing two arcs, both curving the same direction, with PCC marked.
- Curve 1 and Curve 2 with centers at different distances, sharing a tangent line at PCC.
- Station progression: PC₁ → PCC → PT₂ (where PCC = PT₁).
- Deflection angles I₁ and I₂ at respective intersections, summing to total deflection.
Reverse Curves
Formulas
No single formula; analyze as two curves in opposite directions
Meaning
Reverse curve = Curve 1 (R₁, I₁, left) + Curve 2 (R₂, I₂, right), or vice versa.
Watch out
Deflection angles are measured at separate PIs; the curves do NOT share the same PI.
When to use
When two curves turn opposite ways (S-shape); often with a tangent section between.
Offset between parallel tangents = E₁ + E₂ (if radii equal)
Meaning
If R₁ = R₂ and I₁ = I₂, the offset of the final tangent from the initial is 2E.
Watch out
This assumes equal radii and angles; if different, recalculate using geometry of each curve separately.
When to use
Estimating lateral shift when reverse curves are symmetric.
Tangent between reverse curves (if required) has length = T_between
Meaning
Length of straight section connecting PT₁ and PC₂; depends on design standards.
Watch out
Length T_between is NOT computed by formula; it is a design specification (depends on speed, superelevation rate).
When to use
High-speed roads require tangent between reverse curves for superelevation runoff.
Important facts
- Reverse curves turn OPPOSITE directions; do NOT confuse with compound curves (which turn the SAME way).
- If no tangent is between them, PT₁ = PC₂; the curves are back-to-back.
- On high-speed roads, a tangent section is inserted between reverse curves for superelevation runoff (NSCP or AASHTO standards).
- Total deflection = I₁ + I₂, but the curves turn opposite ways.
- Offset between initial and final tangents (lateral shift) = E₁ + E₂ if radii and angles are equal.
- Common on mountain passes and winding roads (e.g., Phil. highways in Cordillera region).
- Design check: Ensure minimum tangent length between reverse curves per road class (NSCP 2015, Table 200 series).
- Field layout: Set out Curve 1 (left), then tangent or immediately Curve 2 (right).
Key definitions
Reverse Curve
Example
Left-turn curve followed by a right-turn curve; looks like the letter 'S' in plan view.
Two circular curves turning OPPOSITE directions (S-shape or back-to-back curves); typically used on mountain or winding roads.
First Curve (Reverse)
Example
Sta PC₁ = 5+000, Sta PT₁ = 5+250.
Initial curve, e.g., turning left; ends at PT₁.
Second Curve (Reverse)
Example
Sta PC₂ = 5+250 (or later if tangent inserted), Sta PT₂ = 5+500.
Subsequent curve, turning opposite direction (e.g., right); starts at PC₂ = PT₁ (if no tangent) or after tangent.
Superelevation Runoff
Example
A 50 m tangent between reverse curves allows the banking to transition from left-tilted to right-tilted.
Transition length where the road is tilted from banking one direction to the opposite; needed between reverse curves at high speed.
Diagrams to know
- Plan view of reverse curve showing S-shape with two curves turning opposite directions.
- Curve 1 (left turn), tangent section (or zero tangent), Curve 2 (right turn).
- Centers of Curve 1 and Curve 2 on opposite sides of the combined path.
- Station progression: PC₁ → PT₁ → [tangent if present] → PC₂ → PT₂.
- Initial tangent, final tangent, and lateral offset illustrated.
Curve Verification & Field Checks
Formulas
Check: LC² = (2R sin(I/2))² = 4R² sin²(I/2)
Meaning
Verify chord length using sine formula.
Watch out
Squaring LC should equal 4R²sin²(I/2); use for verification only.
When to use
Double-check LC calculation in the field or on exam.
Relationship: L_c > LC (always, for I > 0)
Meaning
Arc length is always greater than chord length.
Watch out
If L_c ≈ LC, the angle I is very small (near-straight); if L_c >> LC, angle I is large (sharp curve).
When to use
Quick sanity check: if L_c ≤ LC, you have an error.
Relationship: E ≈ T for small angles; E < T for larger angles
Meaning
External distance and tangent distance are not equal; E decreases relative to T as I increases.
Watch out
For I > 40°, the ratio E/T becomes noticeably different; verify your calculation.
When to use
Sanity check; neither should be zero or vastly different for normal curves.
Station check: Sta PT − Sta PC = L_c (always)
Meaning
Arc length equals the difference in stations between PT and PC.
Watch out
If Sta PT − Sta PC ≠ L_c, you made an error in stationing (often in PI or T calculation).
When to use
Verify stationing is correct.
Important facts
- GOLDEN RULE: Sta PT = Sta PC + L_c. If this is NOT satisfied, recalculate.
- Tangent distance T increases with both R and I; larger curve or bigger angle → longer tangent.
- External distance E is much smaller than T for typical curves.
- Middle ordinate M is the smallest of all elements (M < E < T).
- For I = 0°, all elements → 0; for I → 180°, curve becomes a semicircle.
- In the field, verify at least one element (e.g., measure LC and compare with computed value) to confirm curve layout.
- Degree of curve D increases with sharpness; sharper curves are more sensitive to layout errors.
- Always convert angles to degrees (or radians consistently) before applying formulas.
Diagrams to know
- Side view showing T, E, and M relationships.
- Cross-check table: inputs (R, I) → all outputs (T, L_c, LC, E, M) with verification.
Must remember
- Sta PT = Sta PC + L_c, NOT Sta PC + 2T or Sta PI + T. This is the #1 stationing error.
- Tangent distance T = R tan(I/2); deflection angle I MUST be in degrees (not radians).
- Degree of curve: R = 1145.916/D uses arc definition (20 m standard in Philippines). Chord definition is different.
- Compound curves turn the SAME direction; reverse curves turn OPPOSITE directions (S-shape).
- External distance E = R(sec(I/2) − 1) and middle ordinate M = R(1 − cos(I/2)); they are NOT the same, and E > M always.
- Long chord LC = 2R sin(I/2); arc length L_c = πRI/180. LC < L_c always (except when I = 0).
- PC is located T units BEFORE PI (subtract T); PT is located L_c units after PC (add L_c).
- At a reverse curve, if no tangent is inserted, PT₁ = PC₂ (curves are back-to-back). High-speed roads require a tangent for superelevation.
- Formula verification check: Sta PT − Sta PC = L_c (always true if calculations are correct).
- For compound curves, treat as two independent simple curves; no single combined formula (analyze Curve 1 and Curve 2 separately).
Last minute tips
- ANGLE UNITS: Always ensure I (deflection angle) is in DEGREES when using T = R tan(I/2) and L_c = πRI/180. If given in radians, convert (multiply by 180/π).
- STATIONING TRAP: The most common error is computing Sta PT incorrectly. Write the formula Sta PT = Sta PC + L_c on your answer sheet FIRST to avoid the mistake of adding 2T or using T − L_c.
- ARC vs CHORD: When given a degree of curve D, confirm the definition (arc or chord). In the Philippines, assume ARC definition (20 m standard): R = 1145.916/D. If exam specifies chord, use sin(D/2) = 10/R.
- COMPOUND vs REVERSE: Compound = same direction, PCC is the common point. Reverse = opposite directions (S-shape), often with a tangent between. If problem says 'two curves turning left, then right,' it's a REVERSE curve.
- QUICK SANITY CHECKS: (1) L_c > LC always. (2) E < T always. (3) M < E always. (4) T > 0, E > 0, M > 0 for any I > 0. (5) All should decrease as I → 0. If any fail, recalculate.
Comparison tables
Simple vs Compound vs Reverse Curves
| Feature | Simple Curve | Compound Curve | Reverse Curve |
|---|---|---|---|
| Direction of turn | Single arc (one direction) | Both curves turn SAME direction | Curves turn OPPOSITE directions |
| Number of radii | One radius (R) | Two radii (R₁, R₂) | Two radii (R₁, R₂) |
| Number of PIs | One PI | Two PIs (at separate locations) | Two PIs (at separate locations) |
| Common point | N/A | PCC (point of compound curvature); PCC = PT₁ = PC₂ | PT₁ and PC₂ may be separate (tangent between) or back-to-back |
| Tangent between curves | Not applicable | Usually no tangent; curves are continuous | May have tangent section (required at high speed per NSCP) |
| Total deflection | Single angle I | I_total = I₁ + I₂ | I_total = I₁ + I₂ (but opposite senses) |
| Plan view shape | Single arc | Two arcs, same curvature sense | S-shape or back-to-back arcs |
| Typical use | Most common; highway/rail curves | Transition from sharp to gentle curve; entrance ramps | Mountain roads, winding passes; superelevation transition needed |
Formula Selector: Which Formula When?
| Given Information | Find | Formula(s) | Watch Out |
|---|---|---|---|
| R and I | T, L_c, LC, E, M | T = R tan(I/2); L_c = πRI/180; LC = 2R sin(I/2); E = R(sec(I/2)−1); M = R(1−cos(I/2)) | I in degrees. All five formulas are independent and give all major elements. |
| D and I | R, then all elements | R = 1145.916/D; then apply above formulas | D in degrees (arc definition). Convert D to R first. |
| Sta PI, R, I | Sta PC and Sta PT | Compute T and L_c; Sta PC = Sta PI − T; Sta PT = Sta PC + L_c | Always compute T first, then Sta PC, then L_c, then Sta PT. Never Sta PT = Sta PI + (T − L_c). |
| Sta PC and L_c | Sta PT | Sta PT = Sta PC + L_c | Simplest check: Sta PT − Sta PC should equal L_c exactly. |
| R, and need D | D | D = 1145.916/R | Arc definition (20 m standard in PH). Gives D in degrees. |
| Need to verify a curve in field | Check LC or E or M by measurement | Measure one element; compute from R and I; compare | Best to measure LC (long chord) in field; compare with 2R sin(I/2). |
Common Angle Values & Trig Functions (Quick Ref)
| Angle (degrees) | sin(θ) | cos(θ) | tan(θ) | tan(θ/2) |
|---|---|---|---|---|
| 15° | 0.2588 | 0.9659 | 0.2679 | 0.2679 |
| 20° | 0.3420 | 0.9397 | 0.3640 | 0.3640 |
| 25° | 0.4226 | 0.9063 | 0.4663 | 0.4663 |
| 30° | 0.5000 | 0.8660 | 0.5774 | 0.2679 |
| 35° | 0.5736 | 0.8192 | 0.7002 | 0.3640 |
Diagrams
Simple Curve Calculation Flowchart — Step-by-Step Solution Path
This flowchart guides you through the correct sequence of calculations for a simple curve. Start with R and I, compute all five major elements (T, L_c, LC, E, M), then proceed to stationing if required. The verification step at the end ensures your PT station is correct.
Horizontal Curves — Concept Overview & Relationships
This mind map organizes the entire chapter into conceptual groups: simple curves (elements, stationing, degree), compound curves (same-direction chains), reverse curves (opposite-direction S-curves), and verification techniques. Use this to review the hierarchy and relationships between topics.
Curve Type Identification — Simple vs Compound vs Reverse
Use this decision tree to identify whether you're solving a simple, compound, or reverse curve problem. The key discriminators are: (1) number of radii (one vs two), and (2) direction (same vs opposite). Once identified, follow the appropriate solution path.
Typical Exam Question Breakdown — Horizontal Curves
This pie chart shows the typical distribution of exam questions on horizontal curves. Over one-third focus on computing curve elements (T, L_c, E, M). One-quarter test stationing (PC and PT calculation). Together, these two topics account for 60% of exam content. The remaining 40% covers degree-of-curve conversion, identifying curve types, and field verification.