CELE Surveying (Geomatics) — Vertical (Parabolic) CurvesConcept Map

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

Concept

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

To

Elevation equation: y = elev_PC + g₁·x + (r/2)·x²

From

Constant rate of grade change (r)

Strength

strong

Relationship

r is the core parameter that defines the curvature in the elevation equation

To

Negative rate of grade change (r < 0)

From

Crest curves (g₁ > g₂)

Strength

strong

Relationship

Crest curves always produce negative r values due to the grade decreasing

To

Positive rate of grade change (r > 0)

From

Sag curves (g₁ < g₂)

Strength

strong

Relationship

Sag curves always produce positive r values due to the grade increasing

To

Distance formula: x = g₁·L/(g₁ - g₂)

From

High point (crest curves)

Strength

strong

Relationship

The formula directly locates the summit for crest curves where g₁ > g₂

To

Distance formula: x = g₁·L/(g₁ - g₂)

From

Low point (sag curves)

Strength

strong

Relationship

The same formula locates the valley for sag curves where g₁ < g₂

To

Parabolic offset equation: offset = (g₂ - g₁)·x²/(2L)

From

Vertical offset from tangent

Strength

strong

Relationship

The offset formula quantifies the vertical distance between the parabola and the tangent line

To

Crest curve sight-distance formula

From

Stopping sight distance (S)

Strength

strong

Relationship

S is the primary input parameter determining minimum curve length for crest curves

To

Sight distance formulas

From

Algebraic grade difference (A)

Strength

strong

Relationship

A represents the absolute grade change percentage and directly affects required curve length

To

Sight distance over hump

From

Crest curve design

Strength

strong

Relationship

Crest curves are governed by the need to provide adequate sight distance over the summit

To

Headlight illumination distance

From

Sag curve design

Strength

strong

Relationship

Sag curves at night are governed by headlight beam throw requirements

To

High point elevation and low point elevation

From

Elevation equation: y = elev_PC + g₁·x + (r/2)·x²

Strength

strong

Relationship

The elevation equation is used to calculate the actual height of critical points once their x location is found

To

Distance x from PC along curve

From

PC (Point of Curve) station

Strength

strong

Relationship

The PC serves as the reference point (x = 0) for all distance measurements along the curve

To

All elevation and rate calculations

From

Grade sign convention (+/−)

Strength

strong

Relationship

Correct interpretation of grade signs is essential for all subsequent calculations and determines curve type

To

Maximum mid-curve offset

From

Symmetrical curves

Strength

moderate

Relationship

Symmetrical curves produce maximum offset at the midpoint of the curve length

To

Minimum sight distance requirements

From

Philippine road design guidelines

Strength

moderate

Relationship

Design codes specify the acceptable sight-distance values that govern vertical-curve length

To

Sag curve design

From

Comfort criteria (vertical acceleration)

Strength

moderate

Relationship

Sag curves must provide comfort by limiting the upward acceleration experienced by vehicle occupants

To

Minimum grade in sag curves

From

Drainage requirements

Strength

moderate

Relationship

Sag curves need sufficient grade at the low point to ensure proper drainage

To

Common errors and pitfalls

From

x location from PC (not PI)

Strength

moderate

Relationship

Confusing distance measurement reference is a frequent source of calculation errors

To

Common errors and pitfalls

From

Verifying turning point within curve

Strength

moderate

Relationship

Forgetting to check if 0 ≤ x ≤ L can lead to acceptance of physically impossible solutions

To

Common errors and pitfalls

From

Offset direction (above vs below)

Strength

weak

Relationship

Incorrect offset sign interpretation leads to errors in understanding parabola positioning relative to tangent