GELE Surveying (Geomatics) — Vertical (Parabolic) CurvesRevision Notes
Condensed revision notes for Vertical (Parabolic) Curves, built for the final weeks before the GELE 2026. These are the distilled key points you need when there is no time left for full study notes — just the concepts, formulas, and traps Professional Regulation Commission (PRC) — Board of Geodetic Engineering tests.
Exam context
On the GELE 2026, the Surveying (Geomatics) subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Geodetic Engineering's pattern. Vertical (Parabolic) Curves lands at position 7th out of 9 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Surveying (Geomatics) on a typical GELE paper.
Vertical (Parabolic) Curves - Revision Notes
Vertical curves are parabolic transitions connecting two intersecting road grades in the vertical plane. Because the parabola has a constant rate of grade change, it provides smooth ride quality and predictable sight distances — both critical for highway design. The PRC Civil Engineer Licensure Examination regularly tests: (1) elevation at any point along the curve, (2) location of the high/low point (turning point), (3) vertical offsets from tangent grades, and (4) minimum curve length governed by stopping sight distance. Mastery of the sign convention for grades and the structure of the parabolic elevation equation is the single most important foundation for this topic.
Sections
Formulas
Example
g1 = +0.03, g2 = -0.02, L = 200 m → r = (-0.02 - 0.03)/200 = -0.00025 m⁻¹
Formula
r = (g2 - g1) / L
Variables
r = rate of grade change per metre (m⁻¹); g1 = incoming grade (decimal); g2 = outgoing grade (decimal); L = curve length (m)
Application
Compute r first — it is used in every subsequent calculation. Negative r means crest; positive r means sag.
Exam Tips
- Always write down g1 and g2 as decimals with their correct sign at the start of every problem.
- Sketch the profile immediately — label BVC, PVI, EVC and indicate whether the curve is a crest or sag.
- Confirm: for a crest, g1 > g2; for a sag, g1 < g2.
Key Points
- A vertical curve is always parabolic in standard highway design — NOT circular.
- The parabola provides a constant rate of grade change r = (g2 - g1) / L, which is its defining property.
- BVC (Beginning of Vertical Curve) = PC (Point of Curvature in vertical plane) — the start of the curve.
- EVC (End of Vertical Curve) = PT — the end of the curve.
- PVI (Point of Vertical Intersection) — where the two tangent grades meet; this is analogous to the PI in horizontal curves.
- L = total length of the vertical curve measured horizontally (not along the road surface).
- g1 = incoming grade (positive = uphill, negative = downhill), expressed as a decimal (e.g., +3% = +0.03).
- g2 = outgoing grade (positive = uphill, negative = downhill), expressed as a decimal.
- A crest (summit) curve: g1 > g2 (grade decreases through the curve); rider goes up then down.
- A sag (valley) curve: g1 < g2 (grade increases through the curve); rider goes down then up.
- The PVI lies at horizontal distance L/2 from both the BVC and EVC for a SYMMETRICAL parabolic curve.
- Elevation of PVI = Elev_BVC + g1(L/2).
Definitions
Term
BVC / PC
Definition
Beginning of Vertical Curve — the point where the parabolic curve starts; x = 0 reference for all elevation calculations.
Importance
All distances x are measured FROM the BVC, not from the PVI or EVC.
Term
PVI
Definition
Point of Vertical Intersection — where the two tangent grade lines intersect above or below the road profile.
Importance
Its elevation is needed to compute the mid-curve offset and to verify elevations.
Term
Algebraic Grade Difference (A)
Definition
A = |g1 - g2| expressed in percent. Also written as |g2 - g1| × 100%.
Importance
A is used directly in sight-distance formulas for curve length design.
Term
Rate of Grade Change (r)
Definition
The constant change in grade per unit horizontal length: r = (g2 - g1)/L.
Importance
Determines the shape of the parabola; defines curvature in the vertical plane.
Section Title
1. Fundamental Concepts and Terminology
Common Mistakes
- Using percent values directly in the elevation formula instead of converting to decimals first (e.g., g1 = 3 instead of 0.03).
- Measuring x from the PVI instead of from the BVC.
- Forgetting that L is the HORIZONTAL length, not the slope distance.
- Confusing crest with sag — always check: crest has r < 0; sag has r > 0.
Formulas
Example
Elev_BVC = 100.00 m, g1 = +0.03, r = -0.00025 m⁻¹, x = 80 m: y = 100 + 0.03(80) + (-0.00025/2)(80²) = 100 + 2.40 - 0.80 = 101.60 m
Formula
y = Elev_BVC + g1·x + (r/2)·x²
Variables
y = elevation of curve at point x (m); Elev_BVC = elevation at BVC (m); g1 = incoming grade (decimal); x = horizontal distance from BVC (m); r = (g2 - g1)/L (m⁻¹)
Application
Fundamental equation for ALL elevation problems on a symmetrical parabolic vertical curve. Substitute known values and solve.
Example
g1=+0.03, g2=-0.02, L=200 m, x=80 m: offset = [(-0.02-0.03)/(2×200)]×80² = [-0.05/400]×6400 = -0.80 m (0.80 m below tangent — crest)
Formula
offset = (r/2)·x² = [(g2-g1)/(2L)]·x²
Variables
offset = vertical distance from the BVC tangent to the curve at x (m); positive = above tangent (sag); negative = below tangent (crest)
Application
Used when the problem asks for the offset or correction from the tangent grade rather than the absolute elevation.
Example
g1=+0.03, g2=-0.02, L=200 m: offset_max = (-0.02-0.03)×200/8 = -1.25 m (1.25 m below incoming tangent at x = L/2)
Formula
offset_max = (g2 - g1)·L / 8
Variables
offset_max = maximum vertical offset, occurring at mid-curve (x = L/2) (m)
Application
Quick calculation of the maximum deviation of the parabola from the chord connecting BVC to EVC, and also from the tangent at mid-point (PVI level).
Exam Tips
- Board exam problems almost always give the BVC or PVI elevation; convert to BVC elevation first using Elev_BVC = Elev_PVI - g1·(L/2).
- Always verify: at x = L the formula should return the EVC elevation you can independently check with g2.
- Memorize: mid-curve check → Elev_mid = (Elev_BVC + Elev_EVC)/2 + (g2-g1)L/8.
Key Points
- The primary elevation equation uses x measured horizontally from the BVC.
- The equation has three terms: (1) BVC elevation, (2) linear grade term g1·x, (3) parabolic correction (r/2)·x².
- The parabolic correction is negative for a crest curve (r < 0) and positive for a sag curve (r > 0).
- At x = 0: y = Elev_BVC (check).
- At x = L: y = Elev_BVC + g1·L + (r/2)·L² = Elev_BVC + g1·L + (g2-g1)/2·L = Elev_BVC + (g1+g2)/2·L (check with EVC elevation).
- The tangent elevation at distance x: y_tangent = Elev_BVC + g1·x (elevation ON the incoming grade line).
- Vertical offset from the BVC tangent at x: offset = (r/2)·x² = [(g2-g1)/(2L)]·x².
- The offset is negative (curve is BELOW tangent) for a crest; positive (curve is ABOVE tangent) for a sag.
Definitions
Term
Tangent Elevation
Definition
The elevation on the incoming grade line (extension of g1) at distance x from BVC: y_tan = Elev_BVC + g1·x.
Importance
The difference between tangent elevation and curve elevation is the offset. Boards sometimes ask for the offset, requiring you to compute both.
Term
Mid-Curve Offset
Definition
The vertical distance from the PVI to the curve at the mid-point (x = L/2). Equal in magnitude to (g2 - g1)·L/8 and also to half the difference between PVI elevation and the average of BVC and EVC elevations.
Importance
Used as a check: mid-curve elevation = (Elev_BVC + Elev_EVC)/2 + offset_mid.
Section Title
2. Elevation at Any Point Along the Curve
Common Mistakes
- Applying the formula with x measured from the PVI — always use x FROM the BVC.
- Sign error in r: forgetting that for g1=+3%, g2=-2%, r = (-0.02-0.03)/200 is NEGATIVE.
- Not squaring x in the parabolic term — (r/2)·x², not (r/2)·x.
- Using (g1+g2)/2 as the mid-curve grade — this is the grade AT the midpoint, not the elevation offset.
Formulas
Example
g1=+0.03, g2=-0.02, L=200 m: x_turn = 0.03×200/(0.03-(-0.02)) = 6/0.05 = 120 m from BVC
Formula
x_turn = g1·L / (g1 - g2) = -g1 / r
Variables
x_turn = horizontal distance from BVC to the high/low point (m); g1, g2 in decimal with signs; L = curve length (m); r = rate of grade change (m⁻¹)
Application
Critical formula for every 'find the summit/low point' board question. Use decimals with signs.
Example
Elev_BVC=100.00 m, g1=+0.03, r=-0.00025, x_turn=120 m: Elev = 100 + 0.03(120) + (-0.00025/2)(120²) = 100 + 3.60 - 1.80 = 101.80 m
Formula
Elev_turn = Elev_BVC + g1·x_turn + (r/2)·x_turn²
Variables
Elev_turn = elevation of high or low point (m)
Application
Second step after finding x_turn — always compute the elevation as a separate step.
Exam Tips
- As a quick sanity check for a crest with g1 > 0 and g2 < 0: x_turn = g1·L/(g1-g2). Since both g1 > 0 and (g1-g2) > 0, x_turn is positive — correct.
- If the problem gives the station of the BVC, simply add x_turn to get the station of the high/low point.
- The turning-point elevation is always less than the PVI elevation for a crest, and greater than the PVI elevation for a sag.
Key Points
- The high/low point is where the slope of the parabola equals zero — the turning point.
- Differentiating y with respect to x: dy/dx = g1 + r·x = 0, giving x_turn = -g1/r.
- Substituting r = (g2-g1)/L: x_turn = g1·L / (g1 - g2).
- For a crest curve: g1 > 0 and g2 < 0 are typical, so (g1 - g2) > 0 and x_turn > 0 — the summit is within the curve.
- For a sag curve: g1 < 0 and g2 > 0 are typical; x_turn = g1·L/(g1-g2) still applies — the low point is within the curve.
- VALIDITY CHECK: x_turn must satisfy 0 < x_turn < L. If not, the turning point is outside the curve and there is no summit/low point within the curve.
- Elevation of the turning point: substitute x_turn back into y = Elev_BVC + g1·x_turn + (r/2)·x_turn².
- Alternative check: x_turn from the EVC = L - x_turn = g2·L/(g2 - g1) (distance from EVC to turning point, measured backward).
Definitions
Term
Turning Point
Definition
The point on the parabolic curve where the instantaneous grade equals zero (dy/dx = 0). It is the highest point on a crest curve and the lowest point on a sag curve.
Importance
Boards frequently ask for both its location (distance from BVC) and its elevation. Two-part question, two marks.
Section Title
3. Location of the High Point (Crest) or Low Point (Sag)
Common Mistakes
- Using g1 and g2 in percent instead of decimal in the formula — results in x values 100 times too large.
- Not checking that 0 < x_turn < L — if x_turn falls outside this range, there is no turning point within the curve.
- Computing x from the PVI as if it were the BVC — the formula gives distance FROM the BVC.
- For a sag curve: forgetting that g1 is negative; sign errors give a wrong x_turn.
Formulas
Example
g2-g1 = -0.05, L = 200 m, x = 100 m: offset = [-0.05/(2×200)]×100² = -1.25 m (1.25 m below entry tangent)
Formula
y_offset_from_tangent = (r/2)·x² = [(g2-g1)/(2L)]·x²
Variables
Offset from BVC tangent (m); negative below tangent (crest); positive above tangent (sag)
Application
Direct parabolic offset from the entry tangent grade line.
Example
g2-g1 = -0.05, L = 200 m: offset_mid = (-0.05)(200)/8 = -1.25 m
Formula
offset_mid = (g2 - g1)·L / 8
Variables
offset_mid = vertical offset at mid-curve from the BVC tangent (m); also = vertical distance from chord midpoint to curve at x = L/2
Application
Used as a shortcut to find the mid-curve elevation and as a field-layout check.
Exam Tips
- The '1:4:9 rule': offsets at equal intervals x, 2x, 3x from BVC are in ratio 1:4:9 — useful for field layout and quick checks.
- The parabola bisects PVI-to-chord-midpoint: Elev_curve_at_L/2 = (Elev_PVI + Elev_chord_mid)/2.
Key Points
- The offset from the INCOMING tangent (extension of g1) at any x: offset = (r/2)·x² — parabola lies below the tangent for crest (r < 0).
- The offset from the CHORD (line from BVC to EVC) at any x: y_chord = Elev_BVC + [(Elev_EVC - Elev_BVC)/L]·x; offset_from_chord = y_curve - y_chord.
- The maximum offset from the TANGENT occurs at x = L (at the EVC), not at x = L/2.
- The maximum offset from the CHORD occurs at the mid-point x = L/2 and equals (g2-g1)·L/8.
- The curve bisects the distance from the chord mid-point to the PVI: the parabola is exactly halfway between the chord and the PVI tangent intersection.
- PVI offset from chord mid-point = 2 × (mid-curve offset from chord) = (g2-g1)·L/4.
- These offset properties are used to set out curves in the field by the 'throw from tangent' method.
Definitions
Term
Chord
Definition
The straight line connecting the BVC and EVC elevations.
Importance
The parabola lies between the chord and the tangent lines; knowing offsets from both allows field staking.
Section Title
4. Vertical Offsets and the Chord-Tangent Relationships
Common Mistakes
- Confusing offset from tangent with offset from chord — they are different at most points.
- Thinking the maximum tangent offset is at x = L/2 — it is actually at x = L (at the EVC, offset = (r/2)·L²).
- Not applying the '1:4:9 rule' for parabolic offsets — offsets from the tangent are proportional to the square of the distance: if at x the offset is d, at 2x it is 4d.
Formulas
Example
A = 5%, S = 120 m, h1=1.08 m, h2=0.60 m: L = 5×120²/[200(√1.08+√0.60)²] = 5×14400/[200(1.039+0.775)²] = 72000/[200×3.292] = 72000/658.4 ≈ 109.4 m; since S=120<L=109.4 is FALSE — use Case 2.
Formula
L = A·S² / [200(√h1 + √h2)²] [Crest, S ≤ L]
Variables
L = minimum curve length (m); A = |g1-g2| in percent; S = stopping sight distance (m); h1 = driver eye height (m) ≈ 1.08 m; h2 = object height (m) ≈ 0.60 m
Application
Use when the required sight distance S falls within the curve length (the common design case).
Example
A=5%, S=120 m: L = 2(120) - 200(1.039+0.775)²/5 = 240 - 200(3.292)/5 = 240 - 131.7 = 108.3 m; check: S=120 > L=108.3 ✓ — Case 2 is valid.
Formula
L = 2S - 200(√h1 + √h2)² / A [Crest, S > L]
Variables
Same as above — use when S > L
Application
When the computed L from Case 1 is less than S, switch to this formula.
Example
K_min = 22 m/%, A = 5%: L_min = 22 × 5 = 110 m
Formula
K = L / A (rate of vertical curvature)
Variables
K = rate of vertical curvature (m per percent); L = curve length (m); A = algebraic grade difference (percent)
Application
Design tables list minimum K values for each design speed; multiply by A to get L.
Exam Tips
- When no h1, h2 are given and the formula denominator is provided as 658, memorise: 200(√1.08+√0.60)² ≈ 658.
- If the exam asks for minimum L and gives K, just compute L = K × A (A in percent) — fastest approach.
- Always state which case (S ≤ L or S > L) you are using and verify the condition at the end.
- Sag curve minimum length for comfort: L = A·V²/395 where V is design speed in km/h (from comfort criterion); memorize this as a secondary formula.
Key Points
- Vertical curve length L is often controlled by STOPPING SIGHT DISTANCE (SSD), not just geometry.
- A = |g1 - g2| × 100 = algebraic difference in grades expressed in PERCENT (not decimal) for sight-distance formulas.
- CREST curves: line of sight passes over the hump; restricted by driver eye height h1 ≈ 1.08 m and object height h2 ≈ 0.60 m (AASHTO; local codes may differ).
- SAG curves: headlight illumination at night governs; headlight height ≈ 0.60 m, upward beam angle β ≈ 1°.
- Two cases for CREST (S = stopping sight distance in metres): Case 1 — S ≤ L (sight distance fits within curve); Case 2 — S > L (sight distance extends beyond both ends of curve).
- For Case 1 (S ≤ L) on a crest: L = A·S² / [200(√h1 + √h2)²]; with h1=1.08 m and h2=0.60 m: denominator ≈ 200(1.039+0.775)² ≈ 200(3.29) ≈ 658.
- For Case 2 (S > L) on a crest: L = 2S - 200(√h1+√h2)²/A.
- Always CHECK which case applies after computing L — if the assumed case gives a contradiction, use the other formula.
- PRC board exams typically give the formula explicitly or use K = L/A (the rate of vertical curvature) where K = S²/658 for crest and K = S²/400(headlight formula) for sag.
- K value: once K is found, L = K·A (A in percent). K tables are provided in highway design references.
Definitions
Term
Stopping Sight Distance (SSD)
Definition
The minimum distance required for a driver to see an obstacle and stop safely. It governs minimum vertical curve length.
Importance
PRC board problems frequently specify SSD and ask for minimum L — apply the correct crest or sag formula.
Term
Algebraic Grade Difference (A)
Definition
A = |g1 - g2| in PERCENT (not decimal). If g1 = +3% and g2 = -2%, then A = |3-(-2)| = 5%.
Importance
CRITICAL: sight-distance formulas use A in PERCENT, while the elevation formula uses g1, g2 as decimals — do not mix.
Section Title
5. Sight Distance and Minimum Curve Length
Common Mistakes
- Using A as a decimal (e.g., 0.05) instead of percent (5%) in the sight-distance formula — results in L 100× too large.
- Assuming Case 1 (S ≤ L) without verifying the condition after computing L.
- Using h1 = h2 = 1.0 m generically — use the values given in the problem; exam problems may specify different heights.
- Forgetting to square the (√h1 + √h2) sum — it is (√h1 + √h2)², not (√h1 + √h2).
Exam Tips
- WORKED EXAMPLE A — Elevation at a given point: A vertical curve has g1=+4%, g2=-2%, L=300 m, Elev_BVC=150.00 m. Find elevation at x=150 m. Solution: r=(-0.02-0.04)/300=-0.0002 m⁻¹; y=150+0.04(150)+(-0.0002/2)(150²)=150+6.00-2.25=153.75 m.
- WORKED EXAMPLE B — Low point of sag: g1=-3%, g2=+2%, L=250 m, Elev_BVC=80.00 m. x_turn=(-0.03)(250)/(-0.03-0.02)=(-7.5)/(-0.05)=150 m from BVC. r=(0.02-(-0.03))/250=+0.0002; Elev_low=80+(-0.03)(150)+(0.0002/2)(150²)=80-4.50+2.25=77.75 m.
- WORKED EXAMPLE C — Minimum crest L: g1=+3%, g2=-2%, A=5%, SSD=150 m, h1=1.08 m, h2=0.60 m. Case 1: L=5×150²/658=5×22500/658=171000/658≈169.3 m; check S=150<L=169.3 ✓. Answer: L_min ≈ 169.3 m.
Key Points
- STEP 1: Identify given data — g1, g2 (with signs), L, Elev_BVC or Elev_PVI, station of BVC.
- STEP 2: Convert grades to decimals with signs if given in percent.
- STEP 3: Compute r = (g2 - g1)/L.
- STEP 4: If Elev_PVI is given, find Elev_BVC = Elev_PVI - g1·(L/2).
- STEP 5: Identify what is asked — elevation at x, turning point location/elevation, offset, or minimum curve length.
- STEP 6: Apply the correct formula and VERIFY the result (check at x=L, check sign of offset).
- Always keep at least 4 decimal places for r to avoid rounding errors in elevations.
Section Title
6. Board-Exam Problem-Solving Strategy and Worked Examples
Common Mistakes
- Not converting PVI elevation to BVC elevation before applying the elevation formula.
- Rounding r too early — carry 5-6 significant figures.
- Mixing up 'distance from BVC' and 'station' — distance = Station_x - Station_BVC.
Connections
- Horizontal Curves (Road Design): Just as horizontal curves use circular or spiral geometry to transition between horizontal alignments, vertical parabolic curves transition between vertical grades. Both topics appear together in road design problems requiring combined horizontal-vertical alignment.
- Route Surveying and Staking: The station-elevation table produced from the vertical curve elevation formula is used directly in earthwork computation (cut-and-fill volumes) and field staking — connecting to volumes by prismoidal formula.
- Earthwork and Mass Haul: The turning point (high/low point) of a vertical curve separates cut sections from fill sections in profile, making it critical for determining mass haul limits.
- Transportation Engineering (Highway Design): Minimum vertical curve lengths are governed by AASHTO design standards and local DPWH standards; sight distance theory links vertical curves to horizontal sight lines, speed design, and traffic safety.
- Trigonometry and Analytic Geometry: The parabola y = ax² + bx + c is a fundamental algebraic curve; the vertical curve elevation equation is a direct application of the parabola in engineering coordinates.
- Differential Calculus: Finding the turning point uses dy/dx = 0 — a direct application of calculus optimization. This connection reinforces the relevance of calculus in engineering surveying.
- Field Surveying Methods: Differential leveling is used to set BVC and EVC elevations in the field; the vertical curve elevations computed analytically must then be set out by the surveying crew using the offset-from-tangent method.
- PRC Board Exam — Surveying Component: Vertical curves consistently appear in the Surveying portion of the Civil Engineer board exam, often combined with staking (stationing), earthwork, or sight distance — requiring candidates to integrate multiple surveying subtopics.
Exam Strategy
For PRC Civil Engineer board exam problems on Vertical Parabolic Curves, use this proven attack sequence: (1) READ and SKETCH — draw the profile with BVC, PVI, EVC labeled; mark g1 and g2 with arrows and correct signs. (2) CONVERT — write g1 and g2 as decimals with signs immediately. (3) FIND BVC ELEVATION — if PVI elevation is given, subtract g1·(L/2); if EVC elevation is given, subtract (g1+g2)/2·L from EVC or compute backward. (4) COMPUTE r = (g2-g1)/L — note sign. (5) APPLY the correct formula: elevation at x uses y = Elev_BVC + g1·x + (r/2)·x²; turning point uses x_turn = g1·L/(g1-g2); minimum length uses the sight-distance formula with A in percent. (6) VERIFY — plug x=0 (must give Elev_BVC), x=L (must give Elev_EVC), and check turning point is within [0, L]. (7) WATCH THE UNITS — L in metres, g in decimal for elevation formula but A in percent for sight-distance formula. COMMON BOARD TRAPS: (a) PVI elevation given instead of BVC — always convert; (b) grades given in percent — always convert to decimal for the elevation equation; (c) asking for station of turning point — add x_turn to station of BVC; (d) sight distance case check — always verify Case 1 or Case 2 applies after computing L. TIME ALLOCATION: A vertical curve problem with 2-3 parts typically takes 4-6 minutes. If you get a complex combined problem, do the geometry parts (elevation, turning point) first to bank marks, then attempt the sight-distance part if time allows.
Quick Review Questions
A parabolic crest curve has g1 = +5% and g2 = -3%, with L = 400 m and Elev_BVC = 200.00 m. What is the elevation at x = 100 m from the BVC?
Step 1: Convert grades — g1 = +0.05, g2 = -0.03. Step 2: r = (-0.03 - 0.05)/400 = -0.08/400 = -0.0002 m⁻¹. Step 3: y = 200 + 0.05(100) + (-0.0002/2)(100²) = 200 + 5.00 + (-0.0001)(10000) = 200 + 5.00 - 1.00 = 204.00 m. [Note: re-check arithmetic: 200 + 5.00 - 1.00 = 204.00 m — answer is 204.00 m. The 204.875 above was in error; correct is 204.00 m as shown.]
For the same curve in Q1 (g1=+5%, g2=-3%, L=400 m, Elev_BVC=200.00 m), find the distance from the BVC to the summit and its elevation.
x_turn = g1·L/(g1-g2) = 0.05×400/(0.05-(-0.03)) = 20/0.08 = 250 m. Check: 0 < 250 < 400 ✓. Elev = 200 + 0.05(250) + (-0.0002/2)(250²) = 200 + 12.50 + (-0.0001)(62500) = 200 + 12.50 - 6.25 = 206.25 m.
A sag vertical curve has g1 = -2% and g2 = +4%, L = 200 m. Where is the low point from the BVC?
x_turn = g1·L/(g1-g2) = (-0.02)(200)/((-0.02)-(+0.04)) = -4/(-0.06) = 66.67 m. Since 0 < 66.67 < 200, the low point lies within the curve. Note: for a sag, the formula still applies — just keep signs consistent.
What is the maximum vertical offset from the entry tangent at the mid-point of a crest curve with g1=+3%, g2=-2%, L=200 m?
offset_mid = (g2-g1)·L/8 = (-0.02-0.03)×200/8 = (-0.05)(200)/8 = -10/8 = -1.25 m. The negative sign means the curve lies 1.25 m BELOW the entry tangent at x = L/2 = 100 m, as expected for a crest.
A crest curve must accommodate a stopping sight distance of S = 130 m. The algebraic grade difference A = 4%. Using h1 = 1.08 m and h2 = 0.60 m (denominator = 658), find the minimum curve length.
Case 1: L = A·S²/658 = 4×130²/658 = 4×16900/658 = 67600/658 ≈ 102.7 m. Check: S=130 < L=102.7? NO (130 > 102.7). So Case 1 condition S ≤ L is violated. Use Case 2: L = 2S - 658/A = 2(130) - 658/4 = 260 - 164.5 = 95.5 m. Check: S=130 > L=95.5 ✓. Therefore L_min = 95.5 m.
The PVI of a vertical curve is at Sta 10+000, elevation 105.50 m. g1 = +2%, g2 = -3%, L = 250 m. Find the elevation at Sta 10+060.
BVC is at Sta 10+000 - 250/2 = Sta 9+875. Elev_BVC = 105.50 - 0.02(125) = 105.50 - 2.50 = 103.00 m. x = Sta10+060 - Sta9+875 = 185 m. r = (-0.03-0.02)/250 = -0.0002. y = 103.00 + 0.02(185) + (-0.0002/2)(185²) = 103.00 + 3.70 + (-0.0001)(34225) = 103.00 + 3.70 - 3.4225 = 103.278 m. [Re-checking: 103.00+3.70-3.4225 = 103.278 m — note the answer shown above of 106.598 m assumed a different BVC elevation; correct answer with these values is 103.278 m. Examinees: always compute Elev_BVC carefully from PVI data.]
What does a negative value of r = (g2 - g1)/L indicate about the type of vertical curve?
r = (g2-g1)/L is negative when g2 < g1, meaning the outgoing grade is less than the incoming grade. This is the definition of a crest curve: the road goes up then down (or decreases its upward slope). A sag curve has g2 > g1, so r > 0.
For a parabolic vertical curve, if the offset from the BVC tangent at x = 50 m is 0.30 m, what is the offset at x = 100 m (same curve, same entry tangent)?
Parabolic offsets from the tangent are proportional to x². If at x=50 the offset is 0.30 m, then at x=100 (which is 2× farther): offset = 0.30 × (100/50)² = 0.30 × 4 = 1.20 m. This is the 1:4:9 rule of parabolic geometry — a powerful shortcut for quick checks.
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