Track geometry

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Triple gauge used in the station yards at Gladstone & Peterborough in South Australia Triple Gauge Australia.jpg
Triple gauge used in the station yards at Gladstone & Peterborough in South Australia

Track geometry is three-dimensional geometry of track layouts and associated measurements used in design, construction and maintenance of railroad tracks. The subject is used in the context of standards, speed limits and other regulations in the areas of track gauge, alignment, elevation, curvature and track surface. [1] Although, the geometry of the tracks is three-dimensional by nature, the standards are usually expressed in two separate layouts for horizontal and vertical.



Horizontal layout

Tangent track in blue with transition spiral in red and curved track in green. Easement curve.svg
Tangent track in blue with transition spiral in red and curved track in green.

Horizontal layout is the track layout on the horizontal plane. This can be thought of as the plan view which is a view of a 3-dimensional track from the position above the track. In track geometry, the horizontal layout involves the layout of three main track types: tangent track (straight line), curved track, and track transition curve (also called transition spiral or spiral) which connects between a tangent and a curved track.

In Australia, there is a special definition for a bend (or a horizontal bend) which is a connection between two tangent tracks at almost 180 degrees (with deviation not more than 1 degree 50 minutes) without an intermediate curve. There is a set of speed limits for the bends separately from normal tangent track. [2]

Vertical layout

Vertical layout is the track layout on the vertical plane. This can be thought of as the elevation view which is the side view of the track to show track elevation. In track geometry, the vertical layout involves concepts such as crosslevel, cant and gradient.

Reference rail

The reference rail is the base rail that is used as a reference point for the measurement. It can vary in different countries. Most countries use one of the rails as the reference rail. For example, the United States uses the reference rail as the line rail which is the east rail of tangent track running north and south, the north rail of tangent track running east and west, the outer rail (the rail that is further away from the center) on curves, or the outside rails in multiple track territory. [3] For Swiss railroad, the reference rail for tangent track is the center line between two rails, but it is the outside rail for curved track. [4]

Track gauge

Track gauge or rail gauge (also known as track gage in the United States [5] ) is the distance between the inner sides (gauge sides) of the heads of the two load bearing rails that make up a single railway line. Each country uses different gauges for different types of trains. However, the 1,435 mm (4 ft 8+12 in) gauge is the basis of 60% of the world's railways.

Transverse elevation


The measurement of crosslevel between two rails Crosslevel in track geometry.svg
The measurement of crosslevel between two rails

Crosslevel (or 'cross level') is the measurement of the difference in elevation (height) between the top surface of the two rails at any point of railroad track. The two points (each at the head of each rail) are measured at by the right angles to the reference rail. Since the rail can slightly move up and down, the measurement should be done under load.

It is said to be zero crosslevel when there is no difference in elevation of both rails. It is said to be reverse crosslevel when the outside rail of curved track has lower elevation than the inside rail. Otherwise, the crosslevel is expressed in the unit of height.

The speed limits are governed by the crosslevel of the track. In tangent track, it is desired to have zero crosslevel. However, the deviation from zero can take place. Many regulations have specification related to speed limits of certain segment of the track based on the crosslevel. [5] [6]

For curved track, most countries use the term cant or superelevation to express the difference in elevation and related regulations.


Warp is the difference in crosslevel of any two points within the specific distance along the track. The warp parameter in the track geometry is used to specify the maximum in the crosslevel difference of the track in any segment (tangents, curves and spirals).

Without the maximum warp parameter, the regulation on crosslevel alone may not be sufficient. Consider rails with a positive crosslevel followed by a negative crosslevel followed by a sequence of alternating positive and negative crosslevels. Although, all of those crosslevels are in permissible parameter, when operating a train along such track, the motion will be rocking left and right. Therefore, the maximum warp parameter is used to prevent the critical harmonic rock-off condition that may result in the trains rocking back and forth and derailing following wheel climb. [6]

In the United States, the specific distance used for measurement to ensure that the difference in crosslevel of the track is within the permissible warp parameter is 62 feet. The design warp is zero for both tangent and curved track. That means, ideally, the crosslevel should not change between any two points within 62 feet. There are some deviations to allow crosslevels along the track to change (such as change for superelevation in curves). Different levels of those deviations from the zero warp specify the speed limits. [5]

The specification that focuses on the rate of change in crosslevels of curved track is contained within the area related to cant gradient.

Longitudinal elevation

Track gradient

d = distance traveled horizontally
Dh = rise
l = slope length
a = angle of inclination Track gradient.svg
d = distance traveled horizontally
Δh = rise
l = slope length
α = angle of inclination

The term track gradient is relative elevation of the two rails along the track. This can be expressed in the distance traveled horizontally for a rise of one unit, or in terms of an angle of inclination or a percentage difference in elevation for a given distance of the track.

The allowable gradients may be based on the ruling gradient which is the maximum gradient over which a tonnage train can be hauled with one locomotive. In some countries, momentum gradient which is a steeper but shorter gradient may be allowed. This is usually when there is a track gradient is connected to a leveled tangent track that is long enough with no signal between them so that train can build momentum to push through steeper grade than it can be without momentum.

In curved track (with or without cant), there will be curve resistance to push the trains through the curve. The allowable gradients may be reduced on curves to compensate for the extra curve resistance. [2] The gradient should be uniform along the track.

Vertical curve

Vertical curve is the curve in vertical layout to connect two track gradients together whether it is for changing from an upgrade to a downgrade (summit), changing from a downgrade to an upgrade (sag or valley), changing in two levels of upgrades or changing in two levels of downgrades.

Some countries do not have specification on the exact geometry of vertical curves beyond general specification on vertical alignment. Australia has specification that the shape of vertical curves should be based on quadratic parabola but the length of a given vertical curve is calculated based on circular curve. [2]


Curve with superelevation of tracks on the Keystone Corridor near Rosemont, PA Amtrak Keystone Corridor Rosemont Curve.jpg
Curve with superelevation of tracks on the Keystone Corridor near Rosemont, PA

In most countries, the measurement of curvature of curved track is expressed in radius. The shorter the radius, the sharper the curve is. For sharper curves, the speed limits are lower to prevent an outward horizontal centrifugal force to overturn the trains by directing its weight toward the outside rail. Cant may be used to allow higher speeds over the same curve.

In the United States, the measurement of curvature is expressed in degree of curvature. This is done by having a chord of 100 feet (30.48 m) connecting to two points on an arc of the reference rail, then drawing radii from the center to each of the chord end points. The angle between the radii lines is the degree of curvature. [7] The degree of curvature is inverse of radius. The larger the degree of curvature, the sharper the curve is. Expressing the curve in this way allows surveyors to use estimation and simpler tools in curve measurement. This can be done by using a 62-foot (18.90 m) string line to be a chord to connect the arc at the gage side of the reference rail. Then at the midpoint of the string line (at the 31st foot), a measurement is taken from the string line to the gauge of the reference rail. The number of inches in that measurement is approximated to be the number of degrees of curvature. [5]

Due to the limitation of how specific train equipment can make a turn at maximum speeds, there is a limitation of minimum curve radius to control the sharpness of all curves along a given route. Although most countries use radius for measurement of curvature, the term maximum degree of curvature is still used outside of the United States such as in India, but with the radius as the unit. [8]


Railroad track spirit level in place indicating 5" of superelevation between the inside and outside rails of a curve along the Keystone Corridor near Narberth, PA. 5 inch track superelevation.jpg
Railroad track spirit level in place indicating 5" of superelevation between the inside and outside rails of a curve along the Keystone Corridor near Narberth, PA.

In curved track, it is usually designed to raise the outer rail, providing a banked turn, thus allowing trains to maneuver through the curve at higher speeds that would otherwise be not possible if the surface was flat or level. It also helps a train steer around a curve, keeping the wheel flanges from pressing the rails, minimizing friction and wear. The measurement of the difference in elevation between the outer rail and the inner rail is called cant in most countries. Sometime the cant is measured in term of angle instead of height difference. [9] In the United States, it is measured in height difference and called crosslevel, even for the curved track.

When the outside rail is at higher elevation than the inside rail, it is called positive cant. This is normally the desired layout for curved track. Most counties achieve the desired level of positive cant by raising the outside rail to that level which is called superelevation. For Swiss railroads, the cant is done by rotating at the track axis (center of the two rails) to have outside rail super elevated (raised) at the half rate of the desired cant and the inside rail under elevated (lowered) at the same half rate of the desired cant.

When the outside rail is at lower elevation than the inside rail, it is called negative cant (or reverse crosslevel in the United States). This is not usually a desired layout but it may be unavoidable in some situations such as curves involving turnouts.

There are regulations which limit the maximum cant. This is to control the unloading of the wheels on the outside rail (high rail), especially at low speeds.

Cant gradient

Cant gradient is the amount by which cant is increased or decreased in a given length of track. The change in cant is required in order to connect a tangent track (no cant) to a curved track (with cant) through a transition curve. The rate of change of cant is used to determine the suitable cant gradient for a given design speed. Track twist may also be used to describe cant gradient which may be expressed in percentage of cant change per length unit. [4] However, in the UK, the term track twist is normally used in the context of cant gradient with higher values which are considered to be faults. [9]

In the United States, the required cant gradient on a transition curve to achieve smooth connection between superelevation of curved track and the zero crosslevel of tangent track is called superelevation runoff. In addition to the runoff specification, the regulations related to allowable rate of change in the cant is also part of the general specification on the rate of change in crosslevel called warp parameter. The warp parameter and superelevation runoff help calculate the required length of the runoff for a transition curve. [6]

Cant deficiency

As described, cant may be used to reduce lateral acceleration on trains traveling on curved track. This is to balance the centrifugal force (force pushing outward the curve) and centripetal force (force pushing inward the curve). At a higher speed, the centrifugal force is higher. On the contrary, higher cant creates the higher centripetal force. The calculation for this assumes a constant train speed on a constant radius curve.

When the speed of the train and the amount of cant are in balance (centrifugal matches centripetal), it is called equilibrium. This would make the components of wheel to rail force normal to the plane of the track having the same in aggregate for the outside rail as for the inside rail. This would also make the passengers in the train not to perceive any lateral acceleration (a push toward sideway).

For a fixed amount of cant, the speed that creates balance is called equilibrium speed. For a constant speed of a running train, the amount of required cant to achieve the balance is called equilibrium cant. [9]

In practice, trains are not running on equilibrium cants at curves. The situation is called unbalance, which can be in one of the two following ways. For a given speed, if the actual cant is less than the equilibrium cant, the amount of cant difference is called cant deficiency. In the other word, it is the amount of missing cant to achieve the balance. On the contrary, for a given speed, if the actual cant is higher than the equilibrium cant, the amount of over cant from the balance is called cant excess.

In a shared track configuration for trains with different operating speeds such as freight and higher-speed rail passenger services, the cant on a curve should be considered for both high and low speeds. The higher-speed trains would experience cant deficiency and the lower-speed trains would experience cant excess. These parameters have a significant effect on curve performance which includes safety, passenger comfort, and wear and tear of equipment and rails. [10]


The term alignment is used in both horizontal and vertical layouts to describe the line uniformity (straightness) of the rails.

The horizontal alignment (or alinement in the United States) is done by using a predefined length of string line (such as 62-foot in the US and 20 meters in Australia [2] ) to measure along the gauge side of the reference rail. It is the distance (in inches or millimeters) from the midpoint of the string line to the gauge of the reference rail. The design horizontal alignment for tangent track is zero (perfect straight line on the horizontal layout). The design horizontal alignment on the curved track in the United States is 1 inch for each degree of curvature. Any other readings indicate deviations.

The vertical alignment (or profile in the United States, but not to be confused with rail profile) is the surface uniformity in the vertical plane. The measurement of uniformity is done using a predefined length of string line (normally the same length used in horizontal alignment) along the track. If the midpoint of the measurement has higher elevation, it is called hump deviation. On the other hand, if the midpoint has lower elevation, it is called dip deviation. [6]

These deviations from design alignment are used as parameters to assign speed limits.

See also

Related Research Articles


In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "y = mx + b" and it can also be found in Todhunter (1888) who wrote it as "y = mx + c".

Railroad switch Allows trains to be guided from one track to another

A railroad switch (AE), turnout, or [set of] points (BE) is a mechanical installation enabling railway trains to be guided from one track to another, such as at a railway junction or where a spur or siding branches off.

Grade (slope)

The grade of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope, where zero indicates horizontality. A larger number indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction in which run is the horizontal distance and rise is the vertical distance.

Track gauge Spacing of the rails on a railway track

In rail transport, track gauge or track gage is the spacing of the rails on a railway track and is measured between the inner faces of the load-bearing rails.

Track (rail transport) Rail infrastructure

The track on a railway or railroad, also known as the permanent way, is the structure consisting of the rails, fasteners, railroad ties and ballast, plus the underlying subgrade. It enables trains to move by providing a dependable surface for their wheels to roll upon. For clarity it is often referred to as railway track or railroad track. Tracks where electric trains or electric trams run are equipped with an electrification system such as an overhead electrical power line or an additional electrified rail.

HO scale

HO or H0 is a rail transport modelling scale using a 1:87 scale. It is the most popular scale of model railway in the world. The rails are spaced 16.5 mm (0.650 in) apart for modelling 1,435 mm standard gauge tracks and trains in HO.

New York City Subway rolling stock Overview of the rolling stock of the New York City Subway

The New York City Subway is a large rapid transit system and has a large fleet of rolling stock. As of November 2016, the New York City Subway has 6418 cars on the roster.

Derailment Form of train incident

Rail Transport transformed the world but brought many accidents as improvements in technology increased speed, load other elements, constantly testing the capacity of the technology to work as designed. A common element train in accident is a derailment which occurs when a rail vehicle such as a train runs off its rails. Although many derailments are minor, all result in temporary disruption of the proper operation of the railway system and they are a potentially seriously hazard.

Defect detector

A defect detector is a device used on railroads to detect axle and signal problems in passing trains. The detectors are normally integrated into the tracks and often include sensors to detect several different kinds of problems that could occur. Defect detectors were one invention which enabled American railroads to eliminate the caboose at the rear of the train, as well as various station agents stationed along the route to detect unsafe conditions. The use of defect detectors has since spread to other overseas railroads.

Track transition curve

A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, in which a straight section changes into a curve. It is designed to prevent sudden changes in lateral acceleration. In plane, the start of the transition of the horizontal curve is at infinite radius, and at the end of the transition, it has the same radius as the curve itself and so forms a very broad spiral. At the same time, in the vertical plane, the outside of the curve is gradually raised until the correct degree of bank is reached.

Tamping machine

A tamping machine or ballast tamper is a machine used to pack the track ballast under railway tracks to make the tracks more durable. Prior to the introduction of mechanical tampers, this task was done by manual labour with the help of beaters. As well as being faster, more accurate, more efficient and less labour-intensive, tamping machines are essential for the use of concrete sleepers since they are too heavy to be lifted by hand. Whilst also available as a plain tamper with no lifting or lining function this article will focus on the multi function machines.

Track geometry car

A track geometry car is an automated track inspection vehicle on a rail transport system used to test several parameters of the track geometry without obstructing normal railroad operations. Some of the parameters generally measured include position, curvature, alignment of the track, smoothness, and the crosslevel of the two rails. The cars use a variety of sensors, measuring systems, and data management systems to create a profile of the track being inspected.

Cant deficiency

The term "cant deficiency" is defined in the context of travel of a rail vehicle at constant speed on a constant radius curve. Cant itself is a British synonym for the superelevation of the curve, that is, the elevation of the outside rail minus the elevation of the inside rail. Cant deficiency is present when a vehicle's speed on a curve is greater than the speed at which the components of wheel to rail force are normal to the plane of the track. In that case, the resultant force exerts the outside rail more than the inside rails, in which it creates lateral acceleration toward outside of the curve. In order to reduce cant deficiency, the speed can be reduced or the superelevation can be increased. The amount of cant deficiency is expressed in term of required superelevation to be added in order to bring the resultant force into balance between the two rails. In the contrary, it is said to be "cant excess" if the resultant force exerts more against the inside rail than the outside rail, for instance, a high superelevation curve with a train traveling at a low speed.

In railroad engineering, curve resistance is a part of train resistance, namely the additional rolling resistance a train must overcome when travelling on a curved section of track. Curve resistance is typically measured in per mille, with the correct physical unit being Newton per kilo-Newton or N/kN. Older texts still use the wrong unit of kilogram-force per tonne or kgf/t, which mixes an (outdated) unit of force and a unit of mass. Sometimes also kg/t was used, which confused the resisting force with a mass.

Minimum railway curve radius

The minimum railway curve radius is the shortest allowable design radius for the centerline of railway tracks under a particular set of conditions. It has an important bearing on construction costs and operating costs and, in combination with superelevation in the case of train tracks, determines the maximum safe speed of a curve. The minimum radius of a curve is one parameter in the design of railway vehicles as well as trams; monorails and automated guideways are also subject to a minimum radius.

Rail speed limits in the United States are regulated by the Federal Railroad Administration. Railroads also implement their own limits and enforce speed limits. Speed restrictions are based on a number of factors including curvature, signaling, track condition, the physical condition of a train, and the presence of grade crossings. Like road speed limits in the United States, speed limits for rail tracks and the trains that run on them use miles per hour (mph).

Geometric design of roads Geometry of road design

The geometric design of roads is the branch of highway engineering concerned with the positioning of the physical elements of the roadway according to standards and constraints. The basic objectives in geometric design are to optimize efficiency and safety while minimizing cost and environmental damage. Geometric design also affects an emerging fifth objective called "livability," which is defined as designing roads to foster broader community goals, including providing access to employment, schools, businesses and residences, accommodate a range of travel modes such as walking, bicycling, transit, and automobiles, and minimizing fuel use, emissions and environmental damage.

A railway or railroad is a track where the vehicle travels over two parallel steel bars, called rails. The rails support and guide the wheels of the vehicles, which are traditionally either trains or trams. Modern light rail is a relatively new innovation which combines aspects of those two modes of transport. However fundamental differences in the track and wheel design are important, especially where trams or light railways and trains have to share a section of track, as sometimes happens in congested areas.

Road curves are irregular bends in roads to bring a graduation change of direction. Similar curves are on railways and canals.

Cant (road/rail)

The cant of a railway track or camber of a road is the rate of change in elevation (height) between the two rails or edges. This is normally greater where the railway or road is curved; raising the outer rail or the outer edge of the road creates a banked turn, thus allowing vehicles to maneuver through the curve at higher speeds than would otherwise be possible if the surface is flat or level.


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