Vertical (Parabolic) Curves - Concept Map
Vertical Parabolic Curves in Road Profile Design
Related concepts
Parabolic Curve Fundamentals
Sub concepts
- Constant rate of grade change (r)
- Parabolic shape properties
- Symmetrical vs asymmetrical curves
- Grade notation (decimal and percentage)
Relationship to central
Foundation — defines the mathematical and geometric basis of vertical curves
Curve Parameters and Classification
Sub concepts
- Initial grade (g₁)
- Final grade (g₂)
- Curve length (L)
- Algebraic grade difference (A)
- Crest curves (g₁ > g₂)
- Sag curves (g₁ < g₂)
Relationship to central
Essential inputs — defines curve type and geometry
Elevation Calculations
Sub concepts
- Elevation at PC (beginning of curve)
- Elevation equation: y = elev_PC + g₁·x + (r/2)·x²
- Elevation at any point x
- Elevation at PT (end of curve)
- Elevation at high/low point
Relationship to central
Core computation — determines point heights along the curve
Critical Point Location
Sub concepts
- High point (crest curves)
- Low point (sag curves)
- Distance formula: x = g₁·L/(g₁ - g₂)
- Turning point where grade = 0
- Within-curve validation (0 ≤ x ≤ L)
Relationship to central
Design requirement — identifies summit or valley position
Vertical Offsets and Tangent Relations
Sub concepts
- Offset from tangent line
- Parabolic offset equation: offset = (g₂ - g₁)·x²/(2L)
- Maximum mid-curve offset
- Offset sign (above/below tangent)
- Relationship to grade change
Relationship to central
Geometric property — relates parabola to entry/exit tangents
Sight Distance Requirements
Sub concepts
- Stopping sight distance (S)
- Crest curve sight-distance formula
- Sag curve sight-distance formula
- Eye height (h₁) and object height (h₂)
- Headlight beam angle and throw
- Algebraic grade difference (A) in percent
Relationship to central
Safety governs — controls minimum curve length
Crest Curve Design
Sub concepts
- Negative rate of grade change (r < 0)
- High point elevation
- Sight distance over hump
- Comfort and drainage considerations
- Safety for overtaking maneuvers
Relationship to central
Specific application — uphill to downhill transition
Sag Curve Design
Sub concepts
- Positive rate of grade change (r > 0)
- Low point elevation
- Headlight illumination distance
- Comfort (centripetal acceleration)
- Drainage requirements
Relationship to central
Specific application — downhill to uphill transition
Station and Alignment Notation
Sub concepts
- PC (Point of Curve) station
- PT (Point of Tangent) station
- PI (Point of Intersection) station
- Distance x from PC along curve
- Stationing along horizontal alignment
Relationship to central
Referencing system — locates curve in road profile
Design Standards and Codes
Sub concepts
- Philippine road design guidelines
- AASHTO design standards
- Minimum sight distance requirements
- Comfort criteria (vertical acceleration)
- Drainage slope minimums
Relationship to central
Regulatory framework — ensures safe and compliant design
Common Errors and Pitfalls
Sub concepts
- Grade sign convention (+/−)
- x location from PC (not PI)
- Verifying turning point within curve
- r sign interpretation
- Offset direction (above vs below)
- Unit conversion (decimal vs percent grades)
Relationship to central
Quality assurance — prevents calculation mistakes
Concept connections
- Constant rate of grade change (r)Elevation equation: y = elev_PC + g₁·x + (r/2)·x²r is the core parameter that defines the curvature in the elevation equation
- Crest curves (g₁ > g₂)Negative rate of grade change (r < 0)Crest curves always produce negative r values due to the grade decreasing
- Sag curves (g₁ < g₂)Positive rate of grade change (r > 0)Sag curves always produce positive r values due to the grade increasing
- High point (crest curves)Distance formula: x = g₁·L/(g₁ - g₂)The formula directly locates the summit for crest curves where g₁ > g₂
- Low point (sag curves)Distance formula: x = g₁·L/(g₁ - g₂)The same formula locates the valley for sag curves where g₁ < g₂
- Vertical offset from tangentParabolic offset equation: offset = (g₂ - g₁)·x²/(2L)The offset formula quantifies the vertical distance between the parabola and the tangent line
- Stopping sight distance (S)Crest curve sight-distance formulaS is the primary input parameter determining minimum curve length for crest curves
- Algebraic grade difference (A)Sight distance formulasA represents the absolute grade change percentage and directly affects required curve length
- Crest curve designSight distance over humpCrest curves are governed by the need to provide adequate sight distance over the summit
- Sag curve designHeadlight illumination distanceSag curves at night are governed by headlight beam throw requirements
- Elevation equation: y = elev_PC + g₁·x + (r/2)·x²High point elevation and low point elevationThe elevation equation is used to calculate the actual height of critical points once their x location is found
- PC (Point of Curve) stationDistance x from PC along curveThe PC serves as the reference point (x = 0) for all distance measurements along the curve
- Grade sign convention (+/−)All elevation and rate calculationsCorrect interpretation of grade signs is essential for all subsequent calculations and determines curve type
- Symmetrical curvesMaximum mid-curve offsetSymmetrical curves produce maximum offset at the midpoint of the curve length
- Philippine road design guidelinesMinimum sight distance requirementsDesign codes specify the acceptable sight-distance values that govern vertical-curve length
- Comfort criteria (vertical acceleration)Sag curve designSag curves must provide comfort by limiting the upward acceleration experienced by vehicle occupants
- Drainage requirementsMinimum grade in sag curvesSag curves need sufficient grade at the low point to ensure proper drainage
- x location from PC (not PI)Common errors and pitfallsConfusing distance measurement reference is a frequent source of calculation errors
- Verifying turning point within curveCommon errors and pitfallsForgetting to check if 0 ≤ x ≤ L can lead to acceptance of physically impossible solutions
- Offset direction (above vs below)Common errors and pitfallsIncorrect offset sign interpretation leads to errors in understanding parabola positioning relative to tangent
Diagrams
Vertical Parabolic Curves — Complete Concept Hierarchy
Comprehensive hierarchical mind map showing all major concept areas for vertical parabolic curves. The map organizes knowledge into 11 primary branches covering fundamentals, parameters, calculations, critical points, geometric properties, safety requirements, and design considerations.
Vertical Curve Design and Calculation Workflow
Step-by-step workflow for designing and calculating vertical parabolic curves. Shows decision points for curve type, calculation sequence for critical points and elevations, sight-distance verification, and iteration if length adjustments are needed.
Problem-Solving Strategy: Vertical Curve Calculations
Diagnostic flowchart guiding students through the problem-solving approach for vertical curve questions. Identifies the type of question, selects the appropriate formula or method, executes calculations with quality checks, and validates reasonableness of results.
Grade Sign Convention and Curve Classification
Decision tree for properly classifying vertical curves based on grade signs and determining the rate of grade change. Emphasizes the algebraic sign convention (+ uphill, − downhill) and the interpretation of the resulting r value.
Sequence of Vertical Curve Analysis Tasks
Sequential interaction showing the logical flow from initial problem inputs through parameter calculations, critical point determination, elevation analysis, sight-distance verification, and final design validation.
Vertical Curve State Transitions in Design Process
State diagram showing the progression of vertical curve design from initial input through parameter determination, calculation stages, validation checks, and potential iteration if sight-distance requirements are not met.