PS Exam Preparation

Comprehensive preparation for the NCEES Principles and Practice of Surveying (PS) exam. 5 modules covering all 5 exam domains with 50 in-depth topics.

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Lesson 6

Route Surveys for Alignments

Learning Objectives

After completing this topic, you should be able to:

Explain alignment stationing conventions and their application
Describe and compute horizontal curve elements (radius, tangent, length, deflection, external, middle ordinate)
Describe and compute vertical curve elements (PVC, PVI, PVT, grades, elevations)
Interpret roadway and utility plan/profile sheets
Describe right-of-way survey requirements for route projects
Explain superelevation, spiral curves, and their purpose
Identify the surveyor's role in route survey projects

Overview

Route surveys support the design, construction, and maintenance of linear infrastructure: highways, railroads, pipelines, transmission lines, canals, and other corridor-type projects. The surveyor establishes the horizontal and vertical alignment, provides control and topography within the corridor, stakes the design for construction, and performs right-of-way surveys to define the land rights needed for the project.

Route surveying involves distinctive computations centered on stationing, horizontal curves, and vertical curves. These calculations are among the most frequently tested topics on the PS exam.

Key Concepts

Alignment Stationing

Stationing is a sequential distance-based reference system along a route alignment:

Station 0+00 is the beginning of the project
Station numbers increase with distance along the alignment
Station 10+00 = 1,000 feet from the beginning (in U.S. practice, using the foot system)
Station 1+000 = 1,000 meters from the beginning (in metric practice)
Points between full stations are referenced by adding the offset (e.g., Sta 12+45.67 = 1,245.67 feet from the beginning)

Stationing conventions:

Stationing follows the centerline of the route
Left and right are defined looking in the direction of increasing stations
Offsets are measured perpendicular to the centerline
Example: "25 feet right of Sta 14+50" means 25 feet to the right of the centerline at a point 1,450 feet along the alignment

Equation stations:

When an alignment is revised and the stationing must be maintained for the portions that did not change, an equation station is used
Example: Sta 25+00 Back = Sta 24+50 Ahead means there is a 50-foot discrepancy at that point
All references downstream use the "ahead" stationing

Common wrong path — reading an equation station as a simple offset. An equation station written as "STA 25+00 BK = 24+50 AH" tells you that two different numbering systems coexist on the alignment: everything back of this point uses the "back" stationing; everything ahead uses the "ahead" stationing. The difference (50 ft in this case) is the amount of alignment shortened or lengthened during a revision. Students who treat this as a simple offset ("just add 50 ft to everything") get wrong answers whenever they cross the equation station. Similarly, distances between points on opposite sides of the equation station cannot be computed by simple subtraction of stations — you have to either (a) convert both to the same stationing system, or (b) compute the physical distance by breaking it at the equation station. Exam questions test this by placing two points on opposite sides of an equation and asking for the distance between them.

Quick retrieval check — try before reading on.

▶An alignment has an equation station of "STA 30+00 BK = 29+75 AH." What is the physical distance between point A at STA 28+50 (on the back side) and point B at STA 31+00 (on the ahead side)?

Physical distance = (distance from A to equation station, back side) + (distance from equation station to B, ahead side).

From A (28+50) to 30+00 BK: 3,000 − 2,850 = 150 ft
From 29+75 AH to B (31+00): 3,100 − 2,975 = 125 ft
Total: 150 + 125 = 275 ft.

If you had naively subtracted (3,100 − 2,850 = 250 ft), you'd be 25 ft short of the true distance — the equation added 25 ft of physical alignment that simple subtraction ignores. (Note: if the equation had been a "removed" 25 ft instead — STA 30+00 BK = 30+25 AH — the physical distance would have been 225 ft, 25 ft less than subtraction suggests. The sign of the alignment change determines whether the naïve subtraction is too long or too short.)

Horizontal Curves

Horizontal curves connect tangent sections of the alignment. Simple circular curves are the most common and most tested.

Elements of a simple circular curve:

Element	Symbol	Definition
Point of Curvature	PC	Beginning of the curve (tangent meets curve)
Point of Tangency	PT	End of the curve (curve meets tangent)
Point of Intersection	PI	Intersection of the two tangent lines
Radius	R	Radius of the circular arc
Deflection Angle	$\Delta$ (I)	Angle between the two tangent lines
Tangent Distance	T	Distance from PC (or PT) to PI
Curve Length	L	Arc length from PC to PT
Chord Length	C	Straight-line distance from PC to PT
External Distance	E	Distance from PI to the midpoint of the curve
Middle Ordinate	M	Distance from midpoint of the chord to midpoint of the curve

Curve formulas:

Formula	Equation
Tangent	T = R x tan(I/2)
Curve Length	L = R x I (with I in radians), or L = (I/360) x 2piR
Chord	C = 2R x sin(I/2)
External	E = R x [1/cos(I/2) - 1]
Middle Ordinate	M = R x [1 - cos(I/2)]
Degree of Curve (arc)	D = 5729.578 / R (feet)
Degree of Curve (chord)	D = 2 x arcsin(50/R)

Both degree-of-curve formulas assume the standard U.S. highway practice of a 100-foot reference distance. The constant 5729.578 = 180/π × 100, and the "50" in the chord formula is half the 100-foot standard chord.

Stationing around curves:

PC station = PI station - T
PT station = PC station + L
Note: PT station is NOT equal to PI station + T (a common error)

Compound and Reverse Curves

Compound curve: Two or more consecutive circular arcs curving in the same direction with different radii. The point where they meet is the Point of Compound Curvature (PCC).

Reverse curve: Two consecutive circular arcs curving in opposite directions. The point where they meet is the Point of Reverse Curvature (PRC). Reverse curves are generally avoided in highway design because of the abrupt change in lateral acceleration.

Spiral (Transition) Curves

Spiral curves provide a gradual transition between a tangent and a circular curve:

The radius varies from infinity (at the tangent) to the circular curve radius
The most common spiral is the Euler spiral (clothoid)
Spirals allow gradual introduction of superelevation and centripetal acceleration
Key elements: spiral length (Ls), spiral angle, and throw distance

Figure PS.5.6 — Vertical Curve Elements

Vertical Curves

Vertical curves connect tangent grades in the profile view. They are typically parabolic curves.

Elements of a vertical curve:

Element	Symbol	Definition
Point of Vertical Curvature	PVC	Beginning of the vertical curve
Point of Vertical Intersection	PVI	Intersection of the two grade lines
Point of Vertical Tangency	PVT	End of the vertical curve
Length of Curve	L	Horizontal distance from PVC to PVT
Grade In	g1	Grade of the approaching tangent (as decimal, e.g., +0.03)
Grade Out	g2	Grade of the departing tangent
Rate of Change	r	r = (g2 - g1) / L

Vertical curve elevation formula:

Elevation at distance x from PVC: Elev(x) = Elev(PVC) + g1(x) + [(g2 - g1) / (2L)] x^2

Or equivalently: Elev(x) = Elev(PVC) + g1(x) + (r/2) x^2

Key stations:

PVC station = PVI station - L/2
PVT station = PVI station + L/2
PVC elevation = PVI elevation - g1 x (L/2)

Types of vertical curves:

Crest curve: g1 is more positive than g2 (concave downward); controls sight distance
Sag curve: g1 is more negative than g2 (concave upward); controls headlight illumination distance

High/low point on a vertical curve: The high point of a crest curve (or low point of a sag curve) occurs at: x = -g1 / r = -g1 x L / (g2 - g1)

This point is critical for drainage design and sight distance analysis.

Superelevation

Superelevation is the banking of a roadway on curves to counteract centrifugal force:

Maximum superelevation rates are set by design standards (typically 4-12%)
Superelevation is introduced gradually through the spiral or transition zone
The combined effect of superelevation and friction keeps vehicles on the road

Right-of-Way Surveys

Route projects require right-of-way (ROW) surveys to:

Establish the existing property boundaries within the corridor
Determine the land area needed for the project
Prepare legal descriptions for acquisition parcels
Identify and locate improvements, structures, and utilities within the ROW
Support appraisal and negotiation for land acquisition
Prepare ROW maps showing existing boundaries and proposed acquisition lines

Roadway and Utility Plans

Typical plan sheet components:

Title sheet with project description and location map
Summary of quantities
Typical cross-sections
Plan/profile sheets (plan view on top, profile view on bottom)
Cross-section sheets
Drainage plans and profiles
Utility relocation plans
Right-of-way maps
Construction detail sheets

Utility surveys along routes:

Locate existing utilities within the corridor (both overhead and underground)
Coordinate utility relocations with utility companies
Stake new utility installations (water, sewer, gas, electric, communications)
Determine conflicts between proposed construction and existing utilities

Exam Tips

Memorize the horizontal curve formulas: T = R tan(I/2), L = R x I (radians), C = 2R sin(I/2)
PT station = PC station + L (arc length), NOT PI station + T -- this is a classic exam trap
For vertical curves, the elevation formula Elev(x) = Elev(PVC) + g1(x) + [(g2-g1)/(2L)]x^2 is essential
The high/low point on a vertical curve occurs at x = -g1L/(g2-g1) from PVC
Degree of curve (arc definition): D = 5729.578/R -- know this conversion
Crest curves control sight distance; sag curves control headlight illumination distance
Stationing increases in one direction; left and right are defined facing the direction of increasing stations
Remember that vertical curve length (L) is measured as horizontal distance, not along the slope
Spiral curves transition from infinite radius (tangent) to the curve radius -- they are used to gradually introduce superelevation

PS Exam Preparation

Module 1: Legal Principles

Module 2: Professional Survey Practices

Module 3: Standards & Specifications

Module 4: Business Practices

Module 5: Areas of Practice