Track geometry

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. Curved track can also be categorized into three types. The first type is simple curve which has the same radius throughout that curved track. The second type is compound curve which comprises two or more simple curves of different radii that have the same direction of curvature. The third type is reverse curve which comprises two or more simple curves that has the opposite direction of curvature (sometime known as "S" curve or serpentine curve).[3][4]

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.[5]

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.

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, North America 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.[6] 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.[7]

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.[8][9]

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.[9]

In North America, 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.[8]

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

• 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 a track gradient connects to a leveled tangent track long enough and with no signal between them such that a train can build momentum to push through a steeper grade than it can without the momentum gained on the leveled tangent track.

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.[5] The gradient should be uniform along the track.

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.[5]

Railroad track spirit level in place indicating 5 in (130 mm) of superelevation between the inside and outside rails of a curve along the Keystone Corridor near Narberth, Pennsylvania

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.[12] In North America, 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 North America). 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 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.[7] 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.[12]

In North America, 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.[9]

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.[12]

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.[13]