Vertical (Parabolic) Curves - Memory Anchors
Memory techniques — mnemonics, analogies, micro-stories, and visual associations — dramatically improve recall by hooking new information onto existing mental structures. For the PRC Civil Engineer Licensure Examination, where Surveying questions on vertical curves test your ability to recall formulas under time pressure, these anchors transform abstract equations into vivid mental pictures you can retrieve in seconds. Work through each anchor slowly the first time, visualise it clearly, then test yourself using the recall triggers. The goal is not memorisation by repetition — it is understanding locked in by imagination.
Memory anchors
A vertical curve is parabolic, not circular
Memory aid
Think of a basketball free-throw arc — the ball follows a perfect parabola, NOT a circular arc. Roads use the same shape for their hills and dips. Engineers chose the parabola because the grade changes at a CONSTANT rate, just like the ball's horizontal speed stays constant during flight. Every time you see a basketball, remember: parabola = constant rate of grade change.
Why it works
The basketball analogy ties an abstract geometric choice to a vivid, culturally familiar sports image, making the 'why parabolic?' question instantly answerable.
Example usage
Exam question asks 'What curve shape is used for vertical alignment?' — Picture a free-throw arc → parabola → constant grade-change rate r = (g₂ − g₁)/L.
Recall trigger
Basketball free-throw
Elevation equation: y = elev_PC + g₁x + (r/2)x²
Memory aid
This is EXACTLY the kinematics equation: s = s₀ + v₀t + ½at². Map it: elevation y ↔ position s; elev_PC ↔ initial position s₀; g₁ ↔ initial velocity v₀; x ↔ time t; r ↔ acceleration a. A car starting at the PC with initial 'speed' g₁ and constant 'acceleration' r — the elevation traces its displacement. If you can write the kinematics equation from Physics class, you can write the elevation equation.
Why it works
Engineering students already have kinematics deeply wired. Mapping elevation onto a familiar Physics formula exploits existing neural pathways, dramatically reducing learning effort.
Example usage
At x = 120 m from PC (elev = 100 m), g₁ = +0.03, r = −0.00025: y = 100 + 0.03(120) + (−0.00025/2)(120²) = 100 + 3.6 − 1.8 = 101.8 m.
Recall trigger
Kinematics: s = s₀ + v₀t + ½at²
Turning-point distance: x = g₁L / (g₁ − g₂)
Memory aid
Think of it as 'First Grade wins the Race': x = (First grade × Length) ÷ (grade Difference). Write it as x = g₁L / (g₁ − g₂). The FIRST grade (g₁) multiplies L on top; the DIFFERENCE of grades goes on the bottom. Chant: 'First times Length over the Difference gives the Distance to the peak.'
Why it works
The chant creates an auditory-rhythmic memory. 'First grade wins the race' hooks the idea that g₁ is the numerator's driver, preventing the common error of putting g₂ on top.
Example usage
g₁ = +3%, g₂ = −2%, L = 200 m → x = (0.03 × 200)/(0.03 − (−0.02)) = 6/0.05 = 120 m from PC.
Recall trigger
'First Grade wins the Race'
Crest curve: g₁ > g₂ (rising then falling)
Memory aid
Visualise the classic Philippine Mayon Volcano profile — a graceful upward slope (g₁ positive, large) followed by a downward slope on the other side (g₂ negative, smaller in magnitude). The CREST is the summit crater. g₁ IS GREATER than g₂ because you go UP first (positive g₁) then DOWN (negative g₂). Mayon = CREST.
Why it works
Mayon Volcano is instantly recognisable to Filipino students. The visual of the volcano summit perfectly encodes the crest shape and the g₁ > g₂ condition.
Example usage
When asked to classify a curve with g₁ = +4%, g₂ = −1%: picture Mayon → g₁ > g₂ → CREST (summit) curve.
Recall trigger
Mayon Volcano summit
Sag curve: g₁ < g₂ (falling then rising)
Memory aid
Picture the bottom of a Philippine rice paddy irrigation canal — water flows DOWN into the depression and then the land rises UP again on the other side. The SAG is the low point (a sag in the road). g₁ is negative (downhill) and g₂ is positive (uphill), so g₁ < g₂. Canal bottom = SAG.
Why it works
The canal image is culturally grounded and physically intuitive. The bowl shape of a sag instantly reminds students that you go DOWN then UP.
Example usage
Given g₁ = −4%, g₂ = +1.5% → picture canal bottom → SAG curve → low point exists inside the curve.
Recall trigger
Rice-paddy canal bottom
Mid-curve offset (maximum tangent offset): |A × L / 8|
Memory aid
Rhyme: 'A times L over EIGHT — that's the offset at the middle, mate!' where A = |g₂ − g₁| (algebraic difference as a decimal). The maximum vertical distance between the road parabola and the entry tangent occurs right at the midpoint of the curve and equals AL/8. Eight rhymes with 'great' — it is the GREAT mid-point offset.
Why it works
The rhyme creates an auditory hook. The number 8 is also visually symmetrical, mirroring the fact that the maximum offset is at the symmetric mid-point.
Example usage
A = |−0.02 − 0.03| = 0.05, L = 200 m → offset = (0.05 × 200)/8 = 1.25 m below the tangent (crest).
Recall trigger
Rhyme: 'A times L over EIGHT'
Check that the turning point lies within the curve (0 ≤ x ≤ L)
Memory aid
Story: Engr. Reyes calculated the summit at x = 250 m but the curve length was only L = 200 m. He proudly submitted the answer and failed the board exam. The examiner wrote in red: 'The summit escaped the curve!' Remember Engr. Reyes — ALWAYS check 0 ≤ x ≤ L after computing the turning-point distance. If the result falls outside this range, the true turning point does NOT exist within the designed curve.
Why it works
A cautionary story with a named character creates emotional engagement. Failure consequences anchor the importance of the check permanently.
Example usage
After computing x = g₁L/(g₁ − g₂), immediately verify: Is 0 ≤ x ≤ L? If not, the turning point is outside the curve — do not report it as existing.
Recall trigger
'The summit escaped the curve!' — Engr. Reyes's mistake
Sign convention: grades positive (+) going uphill, negative (−) going downhill
Memory aid
Think of your bank account: depositing money = positive = going UPHILL (gaining elevation). Withdrawing money = negative = going DOWNHILL (losing elevation). A grade of +3% means the road 'deposits' 3 m of elevation per 100 m horizontal. A grade of −2% 'withdraws' 2 m per 100 m. Never mix up the signs — it is like confusing a deposit with a withdrawal: your account (elevation) will be totally wrong.
Why it works
Financial analogies resonate deeply because money is emotionally significant. The deposit/withdrawal framing makes sign errors feel as costly as banking mistakes.
Example usage
Road going uphill at 3% → g₁ = +0.03; next tangent going downhill at 2% → g₂ = −0.02. Never enter them as the same sign.
Recall trigger
Bank deposit (up) / withdrawal (down)
r is negative for a crest, positive for a sag
Memory aid
Draw a smiley face ☺ and a sad face ☹ in your head. SMILEY face = SAG curve (curves upward at the ends) = POSITIVE r (things get better, r > 0). SAD/FROWNY face = CREST curve (curves downward at the ends) = NEGATIVE r (frowning downward, r < 0). Smile = sag = positive. Frown = crest = negative.
Why it works
Emotional facial cues are processed in a different part of the brain, making them extremely sticky memory anchors. The visual symmetry of the smile/frown shapes matches the curve shapes.
Example usage
Given g₁ = +3%, g₂ = −2%: grade decreases → frowny face → CREST → r must be negative. Check: r = (−0.02 − 0.03)/200 = −0.00025 ✓ negative.
Recall trigger
Smiley ☺ = sag (+r); Frowny ☹ = crest (−r)
Crest sight distance — limited by line of sight over the hump
Memory aid
Story: Two jeepneys are approaching each other over a hump in a mountain road in Benguet. Neither driver can see the other — the crest is blocking their line of sight like a wall. The road designer's job is to make the curve LONG enough so that drivers can see far enough to stop safely. This is stopping sight distance S. On a CREST curve, visibility is the problem — the hump blocks your eyes.
Why it works
The Benguet mountain road scenario is relatable to Filipino students. A near-collision story triggers the emotional importance of sight distance design.
Example usage
Exam asks why crest curve length is governed by sight distance → picture the jeepney scenario → line of sight over hump is the critical constraint.
Recall trigger
Two jeepneys on a Benguet mountain hump
Sag curve — limited by headlight throw at night
Memory aid
Imagine driving into a SAG (valley) at night on NLEX. Your headlights beam FORWARD and slightly downward. At the bottom of the sag, the road curves UPWARD ahead — your headlights illuminate only a short stretch of road before hitting the rising pavement. The SHORTER the curve, the further up the road curves away from your headlight beam. A longer sag curve = gentler rise = headlights illuminate more road ahead = safer night driving.
Why it works
The NLEX night driving scenario is vivid and personally experienced by many Filipino engineers. The physical mechanism of headlight throw is intuitive once visualised.
Example usage
Exam asks what controls sag curve length → picture NLEX at night → headlight beam throw is the limiting factor for sag curves.
Recall trigger
NLEX night driving into a valley
Algebraic grade difference A = |g₁ − g₂| in sight-distance formulas
Memory aid
A is for ABSOLUTE change in grade — always take the absolute value. 'A is Always Absolute.' In board exams, grades are given as percentages; when plugging into the sight-distance formula, use the PERCENTAGE values directly (e.g., A = 5, not 0.05) as the standard code formula is calibrated for A in percent. Remember: 'A% = Absolute percent change.'
Why it works
The alliteration 'A is Always Absolute' creates an auditory memory hook, and the reminder about percent units prevents a very common calculation error.
Example usage
g₁ = +3%, g₂ = −2% → A = |3 − (−2)| = 5% → plug A = 5 (not 0.05) into L = AS²/[200(√h₁ + √h₂)²].
Recall trigger
'A is Always Absolute (and in percent!)'
x is measured from the PC (BVC), NOT from the PI (PVI)
Memory aid
Story: During the board exam, Maria measured x from the PI (the vertex of the grades) instead of the PC. She got a beautiful answer — 40 metres off. The answer key said x from PC. The PI is just the 'meeting point of the tangents' — it is not on the curve. The curve STARTS at the PC (Beginning of Vertical Curve, BVC). Maria learned the hard way: 'START from the START — measure x from PC.'
Why it works
The story of a board exam mistake with a named, relatable character drives home the importance of the correct reference point with emotional weight.
Example usage
Given station of PC = Sta 2+000, PI at Sta 2+100: to find elevation at Sta 2+060, compute x = 2+060 − 2+000 = 60 m (from PC, not PI).
Recall trigger
'START from the START — x from PC'
Offset direction: BELOW the entry tangent for a crest, ABOVE for a sag
Memory aid
Visualise a hammock (sag) hanging between two trees — the hammock hangs BELOW the straight rope that would connect the trees. Now flip it: a speed bump (crest) rises ABOVE the flat road. HAMMOCK = SAG = offset is ABOVE the chord (the parabola sits above the tangents at mid-curve). SPEED BUMP = CREST = parabola dips BELOW the entry tangent at mid-curve. The road is always closer to the driver than the tangent would be for a crest.
Why it works
Hammock and speed bump are everyday Filipino experiences. The physical intuition about which direction the road deviates from the tangent becomes unforgettable.
Example usage
Crest curve: mid-curve offset = AL/8 = 1.25 m — this is 1.25 m BELOW the entry tangent elevation at mid-curve.
Recall trigger
Hammock (sag, above tangent) vs Speed bump (crest, below tangent)
The parabolic property: offsets from the tangent are proportional to x²
Memory aid
Think of doubling your running speed: if you double your speed, your kinetic energy QUADRUPLES (KE = ½mv²). Parabolic offsets work the same way — double your distance from the PC and the offset grows by FOUR times (2² = 4). Triple the distance → nine times the offset (3² = 9). This x-squared relationship IS the parabola. Every time you think 'parabola,' think 'x-squared growth,' like kinetic energy.
Why it works
Physics students are deeply familiar with the v² relationship. Mapping the parabolic offset onto kinetic energy reinforces both concepts mutually.
Example usage
If the offset at x = 50 m is 0.3 m, at x = 100 m the offset is 0.3 × (100/50)² = 0.3 × 4 = 1.2 m.
Recall trigger
Kinetic energy doubling rule (v² relationship)
Turning-point alternate form: x = −g₁ / r
Memory aid
Remember it as 'NEGATIVE g₁ over r' — x = −g₁/r. This comes from setting dy/dx = 0: d/dx[g₁x + (r/2)x²] = g₁ + rx = 0 → x = −g₁/r. Think of it as 'the slope goes to ZERO when we cancel g₁ with the ramp r.' Chant: 'Negative first grade divided by rate equals the distance to the gate (turning point).'
Why it works
The chant provides an auditory hook, and showing the calculus derivation briefly satisfies analytical thinkers, making the formula feel earned rather than arbitrary.
Example usage
g₁ = +0.03, r = −0.00025 → x = −(0.03)/(−0.00025) = 120 m ✓ Same as the main formula.
Recall trigger
'Negative first grade over rate'
L/2 is the distance from PC to PVI (PI) for a symmetrical curve
Memory aid
A symmetrical vertical curve is like a perfectly balanced seesaw: the fulcrum (PI) is EXACTLY at the middle. The PC is at one end, the PT is at the other. The PI is L/2 from the PC and L/2 from the PT — always. If someone gives you the PI station, the PC station is (PI station − L/2) and the PT station is (PI station + L/2). Think: Seesaw = Symmetry = PI at centre.
Why it works
The seesaw is a universal childhood experience. Symmetry is the key property being encoded, and the seesaw pivot perfectly represents the PI.
Example usage
PI at Sta 3+100, L = 200 m → PC = Sta 3+100 − 100 = Sta 3+000; PT = Sta 3+100 + 100 = Sta 3+200.
Recall trigger
Balanced seesaw with PI as the pivot
Sight-distance formula for crest: L = AS² / [200(√h₁ + √h₂)²] when S < L
Memory aid
Break the formula into three CHUNKS: CHUNK 1 — Numerator: A × S² (grade difference times distance squared). CHUNK 2 — Denominator magic number: 200. CHUNK 3 — Height factor: (√h₁ + √h₂) squared. Say it rhythmically: 'A-S-squared over TWO-HUNDRED times ROOT-h1 plus ROOT-h2 ALL squared.' The denominator 200(√h₁ + √h₂)² is fixed for a given standard (AASHTO eye height 1.08 m, object height 0.60 m gives denominator ≈ 3.50 — memorise this for Philippine road design).
Why it works
Chunking reduces cognitive load by organising the formula into three memorable pieces. The rhythmic chant creates a musical memory trace.
Example usage
A = 5%, S = 120 m, h₁ = 1.08 m, h₂ = 0.60 m → L = 5(120²)/[200(√1.08 + √0.60)²] = 72000/[200(1.0392+0.7746)²] = 72000/[200(3.291)] = 72000/658.2 ≈ 109.4 m.
Recall trigger
THREE CHUNKS: A·S² | 200 | (√h₁ + √h₂)²
The turning point formula is invalid if g₁ and g₂ have the same sign AND the curve is monotone
Memory aid
Story: A highway rises from +2% to +4% (both positive). A student blindly applies x = g₁L/(g₁ − g₂) = 0.02L/(0.02 − 0.04) = −L. Negative distance — the turning point is BEHIND the PC! The student panics. The lesson: when both grades have the same sign, the parabola is monotone (always rising or always falling) — there is NO turning point WITHIN the curve. The sag or summit is outside. Always check the sign of x and compare to the curve limits.
Why it works
A specific computed negative value creates a vivid 'aha moment.' Students remember the illogical result and the physical explanation behind it.
Example usage
g₁ = +2%, g₂ = +4%, L = 100 m → x = 0.02(100)/(0.02 − 0.04) = −100 m. Outside range 0 to 100 m → no turning point in this curve.
Recall trigger
x came out NEGATIVE — turning point behind the PC
Revision game
I am the shape of every road hill and valley. I am not a circle. My key property is that my slope changes at a CONSTANT rate. What am I?
Answer
A parabola (parabolic vertical curve)
Memory link
A1 — Basketball free-throw arc analogy
I am the 'acceleration' of road elevation. I am negative when the road frowns and positive when the road smiles. I am computed as (g₂ − g₁)/L. What am I called?
Answer
Grade-change rate r
Memory link
A2 — RICE mnemonic; A10 — Smiley/Frowny face
I am EXACTLY the kinematics equation s = s₀ + v₀t + ½at² but for road elevations. Replace s₀ with the PC elevation, v₀ with g₁, t with x, and a with r. What equation am I?
Answer
y = elev_PC + g₁x + (r/2)x²
Memory link
A3 — Kinematics twin analogy
I am the distance from the PC to the highest (or lowest) point of a vertical curve. My formula has the FIRST grade on top multiplied by L, and the DIFFERENCE of grades on the bottom. Who am I?
Answer
Turning-point distance: x = g₁L/(g₁ − g₂)
Memory link
A4 — 'First Grade wins the Race' mnemonic
I am the maximum vertical gap between the road parabola and the entry tangent. I occur at the MIDPOINT of the curve. My formula is surprisingly simple — just A times L divided by a single-digit number. What is that number, and what is the formula?
Answer
8; offset = AL/8
Memory link
A7 — Rhyme: 'A times L over EIGHT — that's the offset, mate!'
Two jeepneys are driving toward each other over a mountain hump in Benguet. They cannot see each other. What design parameter of the vertical curve are they testing?
Answer
Stopping sight distance — the crest curve length must be long enough for drivers to see oncoming vehicles (or objects) in time to stop safely.
Memory link
A11 — Benguet jeepney micro-story
Engr. Reyes calculated x = 250 m for a curve of length L = 200 m and almost submitted it as the summit location. What should he have done before submitting, and what is the correct conclusion?
Answer
He should have verified 0 ≤ x ≤ L. Since 250 > 200 = L, the summit does NOT lie within the designed curve.
Memory link
A8 — 'The summit escaped the curve!' micro-story
In the crest sight-distance formula L = AS²/[200(√h₁ + √h₂)²], is A entered as a decimal (e.g., 0.05) or as a percentage (e.g., 5)? Why does it matter?
Answer
A is entered as a PERCENTAGE value (e.g., A = 5, not 0.05). The constant 200 in the denominator is calibrated for A in percent. Using the decimal form gives an answer 100 times too small.
Memory link
A13 — 'A is Always Absolute (and in percent!)'
Formula mnemonics
r = (g₂ − g₁) / L
Mnemonic
RICE: Rate = (g₂ − g₁) Is Change over Extension (L). 'OUT minus IN over Length.'
When to use
First step in every vertical curve problem. Compute r before finding any elevation or turning-point distance.
What each part means
r = rate of grade change (per metre); g₂ = outgoing (exit) grade (decimal); g₁ = incoming (entry) grade (decimal); L = length of vertical curve (m).
y = elev_PC + g₁x + (r/2)x²
Mnemonic
Kinematics twin: s = s₀ + v₀t + ½at². Map: y↔s, elev_PC↔s₀, g₁↔v₀, x↔t, r↔a. 'Elevation is position; grade is velocity; r is acceleration.'
When to use
Whenever you need the road elevation at any point along the vertical curve. Also used to find the turning-point elevation after computing x.
What each part means
y = elevation at distance x from PC (m); elev_PC = elevation at start of curve (m); g₁ = entry grade (decimal, signed); x = horizontal distance from PC (m); r = grade-change rate (per m, signed).
x_turning = g₁L / (g₁ − g₂) = −g₁ / r
Mnemonic
'First Grade wins the Race': x = (First grade × Length) / (grade Difference). Alternate: 'Negative first over rate gives the gate (turning point).'
When to use
Finding the location of the summit (crest) or sump (sag) — the point of zero slope along the curve.
What each part means
x_turning = distance from PC to turning point (m); g₁ = entry grade (decimal); g₂ = exit grade (decimal); L = curve length (m); r = grade-change rate. Valid only when 0 ≤ x ≤ L.
offset_mid = |g₂ − g₁| × L / 8 = AL/8
Mnemonic
Rhyme: 'A times L over EIGHT — that is the offset, mate!' A = algebraic grade difference (decimal); maximum offset is always at mid-curve (x = L/2).
When to use
Quick check of vertical clearance at mid-curve; also useful in computing the elevation of the PI (vertex) relative to the curve mid-point.
What each part means
offset_mid = maximum vertical distance between the parabola and the entry tangent (m); A = |g₂ − g₁| (absolute grade difference, decimal); L = curve length (m). Offset is below the tangent for crest, above for sag.
L = AS² / [200(√h₁ + √h₂)²] — Crest, S < L case
Mnemonic
THREE CHUNKS: 'A·S² over TWO-HUNDRED times ROOT-HEIGHTS-SQUARED.' Numerator = grade times distance squared. Denominator = 200 × (sum of square roots of eye and object heights)².
When to use
Designing a crest vertical curve for minimum stopping sight distance when the assumption S < L is valid. Check: computed L must be ≥ S for the assumption to hold.
What each part means
L = minimum curve length (m); A = algebraic grade difference in PERCENT (not decimal!); S = stopping sight distance (m); h₁ = driver eye height (m, typically 1.08 m per AASHTO); h₂ = object height (m, typically 0.60 m).
Station_PC = Station_PI − L/2; Station_PT = Station_PI + L/2
Mnemonic
Seesaw symmetry: PI is the pivot. PC is L/2 before the PI; PT is L/2 after. 'Back half to PC, forward half to PT.'
When to use
Setting out the curve in the field or converting between station and x-distance from PC.
What each part means
Station_PI = station of the point of vertical intersection (PVI); L = total curve length; PC (BVC) = beginning of vertical curve; PT (EVC) = end of vertical curve.
Quick recall chains
5-Step Vertical Curve Elevation Solution
Recall test
Without looking, list the 5 steps to find the elevation at any point on a vertical curve. Start with 'P' for PC...
Memory chain
Remember 'PRICE-U': PC first → Rate r → Input x → Calculate y → Express units. Like the price tag on a product: you always START with the product (PC), then compute the rate of tax (r), measure how far you are (x), compute the total price (y), and write the peso sign (units).
Items to remember
- 1. Identify PC station and elevation
- 2. Compute r = (g₂ − g₁) / L
- 3. Compute x = (target station) − (PC station)
- 4. Apply y = elev_PC + g₁x + (r/2)x²
- 5. State the elevation and units
Finding the Summit/Sump Location
Recall test
What are the 5 steps to find the turning-point location and elevation? Begin with 'Classify the curve as...'
Memory chain
CCCES: Classify, Calculate x, Check range, Elevate, State station. 'CLASS A Check Elevations at Stations' — a surveyor's daily mantra.
Items to remember
- 1. Confirm it is a crest (g₁ > g₂) or sag (g₁ < g₂)
- 2. Compute x = g₁L / (g₁ − g₂)
- 3. Check 0 ≤ x ≤ L
- 4. Substitute x into elevation equation
- 5. Report station = PC station + x
Signs and Classifications
Recall test
For a curve with g₁ = +3%, g₂ = −2%: What type? What sign of r? Is offset above or below tangent? What limits its length?
Memory chain
CREST is a FROWN (☹, negative r, below tangent, blocked sight). SAG is a SMILE (☺, positive r, above tangent, dark headlights). Two faces, six facts. Remember the two faces and the six facts flow automatically.
Items to remember
- g₁ > g₂ → Crest (summit) curve → r is negative
- g₁ < g₂ → Sag (valley) curve → r is positive
- Offset below tangent → Crest
- Offset above tangent → Sag
- Crest limited by line of sight → stopping sight distance
- Sag limited by headlight throw → night driving
Sight Distance Formula Components
Recall test
List all 6 components of the crest sight-distance formula. What happens if computed L turns out to be less than S?
Memory chain
Think of a cinema (AS²): A-movie, S-creen, squared excitement. The denominator (200-root-heights) is your 'ticket price' — fixed for a given standard. Always verify your S < L assumption after buying the ticket.
Items to remember
- A = algebraic grade difference in PERCENT
- S = stopping sight distance in metres
- h₁ = driver eye height (1.08 m, AASHTO)
- h₂ = object height (0.60 m, AASHTO)
- Denominator = 200(√h₁ + √h₂)²
- Assumption: S < L must be verified after computing L
Common Board-Exam Pitfalls Checklist
Recall test
Name the 6 pitfalls in order using SMCRAO. Explain each one in one sentence.
Memory chain
Acronym SMCRAO: Signs → Measure from PC → Check range → r-sign → A in percent → Offset direction. 'Smart Mappers Check Routes Accurately Often' — a surveyor's safety motto.
Items to remember
- Use decimal grades with correct signs (+ up, − down)
- Measure x from PC, not from PI
- Check 0 ≤ x ≤ L for turning-point validity
- r is negative for crest, positive for sag
- A is in PERCENT for sight-distance formula
- Offset direction: crest = below tangent; sag = above tangent
Diagrams
Vertical Parabolic Curves — Complete Concept Map
A hierarchical mind map covering all major topics in vertical parabolic curves: definitions, crest vs sag classification, key formulas, turning-point procedure, sight-distance design, and common pitfalls. Use this as a mental roadmap before and after studying.
Problem-Solving Decision Tree for Vertical Curve Board Exam Questions
A decision tree flowchart that guides students through the three most common types of vertical curve board exam questions: (1) finding elevation at a point, (2) locating the summit or sump, and (3) computing minimum curve length for sight distance. Follow the diamond decision nodes to choose the correct formula path.
Crest vs Sag Quick Classification and Properties
A side-by-side comparison flowchart for instantly classifying and recalling all properties of crest versus sag vertical curves based on the relationship between g₁ and g₂.
Distribution of Vertical Curve Question Types in PRC Board Exams
Based on historical PRC Surveying board exam patterns, this pie chart shows the approximate distribution of vertical curve question types. Elevation and summit/sump location together account for 65% of questions — master these two types first.