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This article possibly contains original research. (September 2008) 
The minimum railway curve radius, the shortest design radius, has an important bearing on constructions costs and operating costs and, in combination with superelevation (difference in elevation of the two rails) in the case of train tracks, determines the maximum safe speed of a curve. Superelevation is not a factor on tramway tracks. Minimum radius of curve is one parameter in the design of railway vehicles^{[1]} as well as trams.^{[2]}
The first proper railway was the Liverpool and Manchester Railway which opened in 1830. Like the trams that had preceded it over a hundred years, the L&M had gentle curves and gradients. Amongst other reasons for the gentle curves were the lack of strength of the track, which might have overturned if the curves were too sharp causing derailments. There was no signalling at this time, so drivers had to be able to see ahead to avoid collisions with previous trains. The gentler the curves, the longer the visibility. The earliest rails were made in short lengths of wrought iron, which does not bend like later steel rails that were introduced in the 1850s.
Minimum curve radii for railroads are governed by the speed operated and by the mechanical ability of the rolling stock to adjust to the curvature. In North America, equipment for unlimited interchange between railroad companies are built to accommodate 350foot (106.7 m) radius (16 degrees 26 minutes) or sharper, but normally 410foot (125.0 m) radius (14 degrees) is used as a minimum, as some freight cars are handled by special agreement between railroads that cannot take the sharper curvature. For handling of long freight trains, a minimum 717foot (218.5 m) radius (8 degrees) is preferred.
The sharpest curves tend to be on the narrowest of narrow gauge railways, where almost everything is proportionately smaller.^{[3]}
As the need for more powerful (steam) locomotives grew, the need for more driving wheels on a longer, fixed wheelbase grew too. But long wheel bases are unfriendly to sharp curves. Various type of articulated locomotives Mallet, Garratt, Shay were devised to avoid having to operate multiple locomotives with multiple crews.
More recent diesel and electric locomotives do not have a wheelbase problem and can easily be operated in multiple with a single crew.
Not all couplers can handle very sharp curves. This is particularly true of the European buffer and chain couplers, where the buffers get in the way.
A long heavy freight train, especially those with light and heavy waggons mixed up, may have problems going round very sharp curves, as the drawgear forces may pull intermediate waggons off the rails causing derailments. Solutions might include:
A similar problems occur with harsh changes in gradients (vertical curves).
As a heavy train goes round a bend at speed, the centrifugal force the train exerts on the rails is sufficient to move the actual track, which is only held in place by ballast. To counter this, a cant is used, that is, a height difference between the outside and inside rails on the curve. Ideally the train should be tilted such that resultant (combined) force acts straight "down" through the bottom of the train, so the rails feel little or no sideways force. Some trains are capable of tilting to enhance this effect for passenger comfort.
The cant can't of course be ideal at the same time for both fast passenger trains and slow freight trains.
The relationship between speed and tilt can be calculated mathematically. Specifically, the gravitational and centripetal forces need to be in the ratio 1 : tan θ, where θ is the angle by which the train is tilted due to the cant:
Geometrically, tan θ can be expressed (approximately, for small angles) in terms of the track gauge and the cant:
Combining these gives
which rearranges to give the formula for maximum speed on a curve:
where G is the rail gauge, v is speed in m/s (1 m/s = 3.6 km/h), g is gravitational acceleration (9.8 m/s²), h_{a} is cant, and h_{b} is cant deficiency. The cant deficiency is the amount of additional cant that would be needed to neutralise the centrifugal force, and it is proportional to the curve force.
This table shows examples of curve radii. The values used when building highspeed railways vary, and depends on how much wear and safety desired.
Curve radius  ≤ 33 m/s = 120 km/h  ≤ 56 m/s = 200 km/h  ≤ 69 m/s = 250 km/h  ≤ 83 m/s = 300 km/h  ≤ 97 m/s = 350 km/h  ≤ 111 m/s = 400 km/h 

Cant 160 mm, cant deficiency 100 mm, no tilting trains  630 m  1800 m  2800 m  4000 m  5400 m  7000 m 
Cant 160 mm, cant deficiency 200 mm, with tilting trains  450 m  1300 m  2000 m  no tilting trains planned for these speeds 
A curve should not become a straight all at once, but should gradually increase in radius over time (a distance of around 40 m  80 m for a line with a maximum speed of about 100 km/h). Even worse than curves with no transition are reverse curves with no intervening straight.
The superelevation (aka cant) must also be transitioned.
The higher the speed, the longer the transition.
As a train negotiates a curve, the force it exerts on the track changes. Too tight a 'crest' curve could result in the train leaving the track as it drops away beneath it; too tight a 'trough' and the train will plough downwards into the rails and damage them. More precisely, the support force R exerted by the track on a train as a function of the curve radius r is given by
positive for troughs, negative for crests, where m is the mass of the train and v is the speed in m/s. For passenger comfort the ratio of the gravitational acceleration g to the centripetal acceleration v^{2}/r needs to be kept as small as possible, else passengers will feel large 'changes' in their weight.
As trains cannot climb steep slopes, they have little occasion to go over significant vertical curves, however High Speed 1 (section 2) in the UK has a minimum vertical curve radius of 10000m.^{[5]} High Speed 2, with the higher speed of 400 km/h, stipulates much larger 56000m radii.^{[6]} In both these cases the experienced change in 'weight' is less than 7%.
Gauge  Radius  Location  Notes 

1,435 mm (4 ft 8 ^{1}⁄_{2} in)  7,000 m (22,966 ft)  China  Typical China's highspeed railway network (350 km/h) 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  5,500 m (18,045 ft)  China  Typical China's highspeed railway network (250 km/h~300 km/h) 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  4,000 m (13,123 ft)  China  Typical highspeed railways (300 km/h) 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  3,500 m (11,483 ft)  China  Typical China's highspeed railway network (200~250 km/h) 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  2,000 m (6,562 ft)  China  Typical highspeed railways (200 km/h) 
1,067 mm (3 ft 6 in)  250 m (820 ft)  DRCongo MatadiKinshasa Railway  Deviated 1,067 mm (3 ft 6 in) line. 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  240 m (787 ft)  Border Loop  5,000 long tons (5,100 t; 5,600 short tons)  1,500 m (4,921 ft) 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  200 m (656 ft)  Wollstonecraft Railway Station, Sydney  
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  200 m (656 ft)  Homebush triangle  5,000 long tons (5,100 t; 5,600 short tons)  1,500 m (4,921 ft) 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  190 m (623 ft)  Turkey  Turkey^{[3]} 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  160 m (525 ft)  NSW, Zig Zag  40 km/h 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  100 m (328 ft)  NSW, Batlow, New South Wales  Weight limit: 500 long tons (510 t; 560 short tons) and 300 m (984 ft)  restricted to NSW Z19 class 060 steam locomotives 
1,067 mm (3 ft 6 in)  95 m (311.68 ft)  Newmarket, New Zealand  Extra heavy concrete sleepers ^{[8]} 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  85 m (279 ft)  Windberg Railway (de:Windbergbahn)  (between FreitalBirkigt and DresdenGittersee)  restrictions to wheelbase 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  61 m (200 ft)  London Underground Central line  (between White City and Shepherd's Bush) 
1,067 mm (3 ft 6 in)  60 m (197 ft)  Queensland Railways  
762 mm (2 ft 6 in)  50 m (164 ft)  MatadiKinshasa Railway  original 762 mm (2 ft 6 in) line. 
600 mm (1 ft 11 ^{5}⁄_{8} in)  50 m (164 ft)  Welsh Highland Railway  
1,000 mm (3 ft 3 ^{3}⁄_{8} in)  45 m (148 ft)  Bernina Railway  
600 mm (1 ft 11 ^{5}⁄_{8} in)  40 m (131 ft)  Welsh Highland Railway  on original line at Beddgelert 
762 mm (2 ft 6 in)  40 m (131 ft)  Victorian Narrow Gauge  16 km/h or 10 mph on curves; (32 km/h or 20 mph on straight) 
762 mm (2 ft 6 in)  37.47 m (122.9 ft)  KalkaShimla Railway  or 48 degrees 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  29.00 m (95.14 ft)  New York Subway  ^{[9]} 
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  27.43 m (90 ft)  Chicago 'L'  
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  25 m (82 ft)  Sydney steam tram 040  Hauling 3 trailers 
610 mm (2 ft)  21.2 m (70 ft)  Darjeeling Himalayan Railway  The sharpest curves were originally 13.7 m (45 ft) ^{[10]} 
610 mm (2 ft)  18.25 m (59.9 ft)  Matheran Hill Railway  1 in 20 (5%); 8 km/h or 5 mph on curve; 20 km/h or 12 mph on straight 
1,495 mm (4 ft 10 ^{7}⁄_{8} in)  10.973 m (36.00 ft)  Toronto Streetcar System  
1,067 mm (3 ft 6 in)  10.67 m (35 ft)  Taunton Tramway  
1,435 mm (4 ft 8 ^{1}⁄_{2} in)  10.058 m (33.00 ft)  Boston Green Line  
610 mm (2 ft)  4.9 m (16 ft)  Chicago Tunnel Company  6.1 m (20 ft) in grand unions. 
1,676 mm (5 ft 6 in)  175 m (574 ft)  Indian Railways  the sharpest curve permitted on Broad Gauge 
5 ft 2 ^{1}⁄_{2} in (1,588 mm)  28 ft (8.534 m) in yard, 50 ft (15.240 m) elsewhere^{[11]}  Streetcars in New Orleans 


