Route Surveying and Curves

Route Surveying and Curves

Learning Objectives

Understand the principles of route surveying and its applications in civil engineering.
Identify the key elements and geometric properties of simple horizontal curves.
Calculate essential curve components such as tangent distance, length, and external distance.
Apply superelevation concepts to horizontal curve design.
Analyze the characteristics and formulas associated with vertical curves and sight distance.

This topic explores the fundamental concepts and mathematics used to design horizontal and vertical curves, ensuring smooth and safe transitions in transportation alignments.

A specialized branch of surveying focused on the alignment and design of linear structures such as highways, railways, pipelines, and transmission lines. A major component is the geometric design of curves to provide smooth transitions between straight segments.

Horizontal Curves

Horizontal curves connect two intersecting straight lines (tangents) in the horizontal plane to allow vehicles to navigate the change in direction smoothly.

Simple Circular Curves

A simple curve consists of a single circular arc connecting two straights.

Curve Elements

PC (Point of Curvature): The beginning of the curve.
PT (Point of Tangency): The end of the curve.
PI (Point of Intersection): Where the two tangents intersect.
R (Radius): The radius of the circular arc.
$\Delta$ (Deflection Angle or Intersection Angle): The angle by which the forward tangent deflects from the back tangent. Also equals the central angle subtended by the arc.
T (Tangent Distance): Distance from PC to PI, or PI to PT.
L (Length of Curve): The length of the circular arc from PC to PT.
LC (Long Chord): The straight-line distance from PC to PT.
E (External Distance): The distance from PI to the midpoint of the curve.
M (Middle Ordinate): The distance from the midpoint of the curve to the midpoint of the long chord.

Interactive Simulation

Adjust the radius and intersection angle to observe the geometric changes in the tangent and length of a simple horizontal curve.

Simple Horizontal Curve Simulator

Adjust curve parameters to visualize tangent, length, and mid-ordinate changes.

Radius (

R

) (m)200

Intersection Angle (

I

) (°)60.0

Results

T = R \tan(I/2) = 115.47 \text{ m}

L = R \cdot I_{rad} = 209.44 \text{ m}

Mathematical Formulas

Tangent Distance (T)

Calculates the distance from the point of curvature (PC) or point of tangency (PT) to the point of intersection (PI).

T = R \tan \left( \frac{\Delta}{2} \right)

Variables

Symbol	Description	Unit
$T$	Tangent distance	m or ft
$R$	Radius of the circular arc	m or ft
$\Delta$	Deflection angle	degrees

Length of Curve (L)

Computes the arc length from the point of curvature (PC) to the point of tangency (PT).

L = \frac{\pi R \Delta}{180^\circ}

Variables

Symbol	Description	Unit
$L$	Length of curve	m or ft
$R$	Radius of the circular arc	m or ft
$\Delta$	Deflection angle	degrees

Long Chord (LC)

Calculates the straight-line distance from PC to PT.

LC = 2 R \sin \left( \frac{\Delta}{2} \right)

Variables

Symbol	Description	Unit
$LC$	Long chord distance	m or ft
$R$	Radius of the circular arc	m or ft
$\Delta$	Deflection angle	degrees

External Distance (E)

Measures the shortest distance from the point of intersection (PI) to the curve.

E = R \left( \sec \left( \frac{\Delta}{2} \right) - 1 \right)

Variables

Symbol	Description	Unit
$E$	External distance	m or ft
$R$	Radius of the circular arc	m or ft
$\Delta$	Deflection angle	degrees

Middle Ordinate (M)

Measures the distance from the midpoint of the curve to the midpoint of the long chord.

M = R \left( 1 - \cos \left( \frac{\Delta}{2} \right) \right)

Variables

Symbol	Description	Unit
$M$	Middle ordinate	m or ft
$R$	Radius of the circular arc	m or ft
$\Delta$	Deflection angle	degrees

Interactive Curve Calculator

Interactive Simulation

Use the interactive curve calculator below to visualize different horizontal curve parameters.

Horizontal Curve Simulation

Radius (

R

)200 m

Deflection Angle (

\\\\Delta

)60°

Tangent (

T

)115.47 m

Curve Length (

L

)209.44 m

Long Chord (

LC

)200.00 m

External Distance (

E

)30.94 m

Middle Ordinate (

M

)26.79 m

D

The sharpness of the curve can be defined by its Degree of Curve. There are two definitions:

Arc Definition vs. Chord Definition

1. Arc Definition: The central angle subtended by a $20 \, m$ (or $100 \, ft$ ) arc. Standard for highway design. Formula: $D = \frac{1145.916}{R}$ (Metric, $20 \, m$ arc).
2. Chord Definition: The central angle subtended by a $20 \, m$ (or $100 \, ft$ ) chord. Often used in railway design. Formula: $\sin \left( \frac{D}{2} \right) = \frac{10}{R}$ (Metric, $20 \, m$ chord).

Compound, Reverse, and Spiral Curves

Other Curve Types

Compound Curve: Consists of two or more circular arcs of different radii curving in the same direction, joining at a common tangent point (Point of Compound Curvature, PCC). The centers of the curves are on the same side of the alignment.
Reverse Curve: Consists of two circular arcs curving in opposite directions, joining at a common tangent point (Point of Reverse Curvature, PRC). The centers of the curves are on opposite sides. A short straight section is usually required between them to allow for superelevation transition.
Spiral (Transition) Curve: A curve with a continuously changing radius. It is used to connect a straight tangent to a circular curve, allowing for a gradual introduction of centrifugal force and superelevation for passenger comfort and safety. The most common type is the clothoid spiral.

e

Also known as banking, superelevation is the raising of the outer edge of a curved roadway relative to the inner edge. It creates an inward horizontal force component that helps counteract the outward centrifugal force acting on a vehicle, reducing the reliance on tire friction and improving passenger comfort.

Superelevation Formula

Relates superelevation, side friction, velocity, and curve radius.

e + f = \frac{v^2}{gR}

Variables

Symbol	Description	Unit
$e$	Rate of superelevation	m/m or ft/ft
$f$	Coefficient of side friction between tires and pavement	unitless
$v$	Design velocity of the vehicle	m/s or ft/s
$g$	Acceleration due to gravity	$m/s^2 or ft/s^2$
$R$	Radius of the curve	m or ft

Vertical Curves

Vertical curves are used in highway and railway profiles to provide a smooth transition between intersecting grade lines (tangents). They are almost always parabolic rather than circular because a parabola provides a constant rate of change of grade.

Types of Vertical Curves

Crest and Sag Curves

Crest Vertical Curves: Curves that connect an ascending grade with a descending grade, or two grades where the second is less positive than the first. The curve opens downward. Critical design factor is Sight Distance.
Sag Vertical Curves: Curves that connect a descending grade with an ascending grade, or two grades where the second is more positive than the first. The curve opens upward. Critical design factors are headlight sight distance and drainage.

Elements and Formulas

Vertical Curve Elements

BVC (Beginning of Vertical Curve) / PVC (Point of Vertical Curvature): The start of the curve.
EVC (End of Vertical Curve) / PVT (Point of Vertical Tangency): The end of the curve.
PVI (Point of Vertical Intersection): Where the two grade lines intersect.
$g_1, g_2$ : The grades (in percent) of the back and forward tangents, respectively.
$L$ : The horizontal length of the curve.
$r$ : The rate of change of grade per station. $r = \frac{g_2 - g_1}{L}$

Equation of the Parabola: The elevation $y$ at any distance $x$ from the BVC is given by:

Parabolic Curve Equation

Computes the elevation on a vertical curve at any horizontal distance x from the beginning of the curve.

y = y_{BVC} + g_1 x + \frac{r x^2}{2}

Variables

Symbol	Description	Unit
$y$	Elevation on the curve at distance x	m or ft
$y_{BVC}$	Elevation at the Beginning of Vertical Curve	m or ft
$g_1$	Grade of the back tangent (must be a decimal if x is in meters, or percent if x is in stations)	decimal or %
$x$	Horizontal distance from the BVC	m or stations
$r$	Rate of change of grade per station (decimal or % depending on x)	decimal/m or %/station

Sight Distance on Vertical Curves

Sight Distance Consideration

The primary criterion for determining the length of a vertical curve is ensuring adequate Stopping Sight Distance (SSD). The curve must be long enough so that a driver can see an object over a crest (or within headlights in a sag) and stop safely. There is a direct relationship between vertical curve length ( $L$ ) and SSD ( $S$ ).

If $S < L$ :

Crest Curve Sight Distance

Determines the required length of a crest vertical curve based on sight distance when S < L.

L = \frac{A S^2}{100(\sqrt{2h_1} + \sqrt{2h_2})^2}

Variables

Symbol	Description	Unit
$L$	Length of the vertical curve	m or ft
$A$	Algebraic difference in grades \|g_1 - g_2\|	%
$S$	Stopping sight distance	m or ft
$h_1$	Height of the driver's eye	m or ft
$h_2$	Height of the object	m or ft

Key Takeaways

Horizontal Curves: Provide smooth direction changes between two tangent lines using circular arcs.
Key Variables: Radius ( $R$ ) and Deflection Angle ( $\Delta$ ) uniquely define the horizontal curve.
Stationing Rule: Always calculate PT stationing as PC + Curve Length ( $L$ ), never PI + Tangent ( $T$ ).
Superelevation: Banks the road to safely counteract centrifugal force ( $e+f = v^2/gR$ ).
Vertical Curves: Provide smooth transitions between grade lines and are parabolic to ensure a constant rate of change of grade.
Sight Distance: The governing factor in choosing the length ( $L$ ) of a vertical curve.

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Learning Objectives

Horizontal Curves

Simple Circular Curves

Curve Elements

Interactive Simulation

Simple Horizontal Curve Simulator

Results

Mathematical Formulas

Tangent Distance (T)

Length of Curve (L)

Long Chord (LC)

External Distance (E)

Middle Ordinate (M)

Interactive Curve Calculator

Interactive Simulation

Horizontal Curve Simulation

Degree of Curve ( $D$ )

Arc Definition vs. Chord Definition

Compound, Reverse, and Spiral Curves

Other Curve Types

Superelevation ( $e$ )

Superelevation Formula

Vertical Curves

Types of Vertical Curves

Crest and Sag Curves

Elements and Formulas

Vertical Curve Elements

Parabolic Curve Equation

Sight Distance on Vertical Curves

Sight Distance Consideration

Crest Curve Sight Distance

Route Surveying and Curves

Learning Objectives

Horizontal Curves

Simple Circular Curves

Curve Elements

Interactive Simulation

Simple Horizontal Curve Simulator

Results

Mathematical Formulas

Tangent Distance (T)

Length of Curve (L)

Long Chord (LC)

External Distance (E)

Middle Ordinate (M)

Interactive Curve Calculator

Interactive Simulation

Horizontal Curve Simulation

Degree of Curve (DDD)

Arc Definition vs. Chord Definition

Compound, Reverse, and Spiral Curves

Other Curve Types

Superelevation (eee)

Superelevation Formula

Vertical Curves

Types of Vertical Curves

Crest and Sag Curves

Elements and Formulas

Vertical Curve Elements

Parabolic Curve Equation

Sight Distance on Vertical Curves

Sight Distance Consideration

Crest Curve Sight Distance

Degree of Curve ( $D$ )

Superelevation ( $e$ )