View source for Minimum railway curve radius

{{Short description|Concept in railway engineering}}
{{Use British English|date=May 2022}}
{{more citations needed|date=December 2019}}
[[File:CTA loop junction.jpg|thumb|upright=1.5|{{convert|90|ft|m|2|adj=on}} radii on the elevated {{Track gauge|ussg|allk=on}} [[Chicago 'L']]. There is no room for longer radii at this [[Junction (rail)|cross junction]] in the northwest corner of [[The Loop (CTA)|the Loop]].]]

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 [[Cant (road/rail)#Rail|superelevation]] (difference in elevation of the two rails) in the case of [[Track (rail transport)|train tracks]], determines the maximum safe speed of a curve. The minimum radius of a curve is one parameter in the design of [[Railroad car|railway vehicles]]<ref>{{cite web|url=http://worldtraderef.com/WTR_site/Rail_Cars/Guide_to_Rail_Cars.asp|title=Guide to Railcars|website=World Trade Ref - Your One Stop Resource for Trade Information |archive-url=https://web.archive.org/web/20111031040725/http://www.worldtraderef.com/WTR_site/Rail_Cars/Guide_to_Rail_Cars.asp |archive-date=2011-10-31}}</ref> as well as [[tram]]s;<ref>{{cite web|url=http://transittoronto.ca/streetcar/4503.shtml|title=The Canadian Light Rail Vehicles (The CLRVs) - Transit Toronto - Content|website=transittoronto.ca}}</ref> [[monorail]]s and [[Automated guideway transit|automated guideways]] are also subject to a minimum radius.

== History ==
The first proper railway was the [[Liverpool and Manchester Railway]], which opened in 1830.  Like the tram roads that had preceded it over a hundred years, the L&M had gentle curves and [[Grade (slope)|gradients]]. Reasons for these gentle curves include the lack of strength of the track, which might have overturned if the curves were too sharp causing derailments. The gentler the curves, the greater the visibility, thus boosting safety via increased situational awareness. The earliest [[Rail profile|rails]] were made in short lengths of [[wrought iron]],{{cn|reason=wrought iron does bend, it is cast iron that is brittle and goes not bend|date=November 2021}} which does not bend like later [[steel]] rails introduced in the 1850s.

== Factors affecting the minimum curve radius ==

Minimum curve radii for railways 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 railway companies is built to accommodate for a  {{convert|288|ft|m|0|adj=on}} radius, but normally a {{convert|410|ft|m|0|adj=on}} radius is used as a minimum, as some freight carriages (freight cars) are handled by special agreement between railways that cannot take the sharper curvature. For the handling of long freight trains, a minimum {{convert|574|ft|m|0|adj=on}} radius is preferred.<ref name="ZieglerResearch">{{cite book|title=The Shasta Route: Connecting Oregon and California by Passenger Rail|last=Ziegler|first=Hans-Joachim|date=2005-10-28|chapter=Results of railway history|chapter-url=http://zierke.com/shasta_route/pages/13history.html|page=13|url=http://zierke.com/shasta_route/index.html|access-date=5 December 2018}}</ref>

The sharpest curves tend to be on the narrowest of [[Narrow gauge railway|narrow gauge]] railways, where almost all the equipment is proportionately smaller.<ref name="Jane">[[Jane's World Railways]] 1995-1996 p728</ref> But standard gauge can also have tight curves, if rolling stocks are built for it, which however removes the standardisation benefit of standard gauge. Tramways can have below {{convert|100|ft|m|0|adj=on}} curve radius.

=== Steam locomotives ===
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 do not cope well with curves of a small radius. Various types of [[articulated locomotive]]s (e.g., [[Mallet locomotive|Mallet]], [[Garratt]], [[Meyer locomotive|Meyer]] & [[Fairlie locomotive|Fairlie]]) were devised to avoid having to operate multiple locomotives with multiple crews.

More recent diesel and electric locomotives do not have a wheelbase problem, as they have flexible [[Bogie|bogies]], and also can easily be operated in multiple with a single crew.

*The [[Tasmanian Government Railways K class]] was
**{{RailGauge|610mm|lk=on}} gauge
**{{convert|99|ft|m|0|abbr=on}} radius curves
*'''Example Garratt'''
**{{RailGauge|1000mm|allk=on}} 
**{{convert|25|kg/m|lb/yd|abbr=on}} [[Rail profile|rails]]
**Main line radius - {{convert|175|m|ft|0|abbr=on}}
**Siding radius - {{convert|84|m|ft|0|abbr=on}} <ref>{{cite web|url=http://www.garrattmaker.com/history.html|title=Metre-Gauge Beyer-Garratt 4-8-4 + 4-8-4|website=www.garrattmaker.com}}</ref>
*'''0-4-0'''
**[[GER Class 209]]
**{{RailGauge|1435mm|allk=on}}

=== Couplings ===
Not all [[Railway coupling|couplers]] can handle very short radii. This is particularly true of the European [[buffer and chain coupler]]s, where the buffers extend the length of the rail car body.  For a line with a maximum speed of {{convert|60|km/h|mph|abbr=on}}, buffer-and-chain couplers increase the minimum radius to around {{convert|150|m|yd ft|0|abbr=on}}. As [[narrow-gauge railway]]s, [[tram]]ways, and [[rapid transit]] systems normally do not interchange with mainline railways, instances of these types of railway in Europe often use bufferless central couplers and build to a tighter standard.

=== Train lengths ===
A long heavy freight train, especially those with wagons of mixed loading, may struggle on short radius curves, as the [[Railway coupling#Draw gear|drawgear]] forces may pull intermediate wagons off the rails.  Common solutions include:
* marshaling light and empty wagons at the rear of the train
* intermediate locomotives, including remotely controlled ones
* easing curves
* reduced speeds
* reduced cant (superelevation), at the expense of fast passenger trains
* more, shorter trains
* equalizing wagon loading (often employed on [[unit train]]s) 
* better driver training
* driving controls that display drawgear forces
*[[Electronically Controlled Pneumatic]] brakes

A similar problem occurs with harsh changes in gradients (vertical curves).

=== Speed and cant===
As a heavy train goes around a bend at speed, the [[reactive centrifugal force]] may cause negative effects: passengers and cargo may experience unpleasant forces, the inside and outside rails will wear unequally, and insufficiently anchored tracks may move.{{Dubious|date=December 2014|Centrifugal_force_is_not_countered_by_cant}} To counter this, a [[cant (road and rail)|cant]] (superelevation) is used. Ideally, the train should be tilted such that [[resultant force]] acts vertically downwards through the bottom of the train, so the wheels, track, train and passengers feel little or no sideways force ("down" and "sideways" are given with respect to the plane of the track and train). Some trains are capable of [[tilting train|tilting]] to enhance this effect for passenger comfort. Because freight and passenger trains tend to move at different speeds, a cant cannot be ideal for both types of rail traffic.

The relationship between speed and tilt can be calculated mathematically. We start with the formula for a balancing [[centripetal force]]: ''θ'' is the angle by which the train is tilted due to the cant, ''r'' is the curve radius in meters, ''v'' is the speed in meters per second, and ''g'' is the [[standard gravity]], approximately equal to 9.81 m/s²:
:<math>\tan\theta=\frac{v^2}{gr}</math>
Rearranging for ''r'' gives:
:<math>r=\frac{v^2}{g\tan\theta}</math>
Geometrically, tan ''θ'' can be expressed (using the [[small-angle approximation]]) in terms of the [[track gauge]] ''G'', the cant ''h<sub>a</sub>'' and [[cant deficiency]] ''h<sub>b</sub>'', all in millimeters:
:<math>\tan\theta\approx\sin\theta=\frac{h_a+h_b}{G}</math>
This approximation for tan ''θ'' gives:
:<math>r=\frac{v^2}{g\frac{h_a+h_b}{G}}=\frac{Gv^2}{g(h_a+h_b)}</math>
This table shows examples of curve radii. The values used when building high-speed railways vary, and depend on desired wear and safety levels.
{| class="wikitable"
|- Class="hintergrundfarbe5"
! '''Curve radius'''
!{{Cvt|33|m/s|km/h mph|disp=br()|abbr=|order=flip}}
!{{Cvt|56|m/s|km/h mph|disp=br()|order=flip|abbr=}}
!{{Cvt|69|m/s|km/h mph|disp=br()|order=flip}}
!{{Cvt|83|m/s|km/h mph|disp=br()|order=flip}}
!{{Cvt|97|m/s|km/h mph|disp=br()|order=flip}}
!{{Cvt|111|m/s|km/h mph|disp=br()|order=flip}}
|-
| Cant {{convert|160|mm|in|abbr=on}},<br />cant deficiency {{convert|100|mm|in|abbr=on}},<br />no [[tilting train]]s
| {{convert|630|m|ft|abbr=on}}
| {{convert|1800|m|ft|abbr=on}}
| {{convert|2800|m|ft|abbr=on}}
| {{convert|4000|m|ft|abbr=on}}
| {{convert|5400|m|ft|abbr=on}}
| {{convert|7000|m|ft|abbr=on}}
|-
| Cant {{convert|160|mm|in|abbr=on}},<br />cant deficiency {{convert|200|mm|in|abbr=on}},<br />with [[tilting train]]s
| {{convert|450|m|ft|abbr=on}} 
| {{convert|1300|m|ft|abbr=on}}
| {{convert|2000|m|ft|abbr=on}}
| colspan=3 | no tilting trains planned for these speeds{{why|date=September 2024}}
|}
Tramways typically do not exhibit cant, due to the low speeds involved.  Instead, they use [[Tramway track#Grooved rail|the outer grooves of rails as a guide]] in tight curves.

== Transition curves ==

{{Main|Track transition curve}}

A curve should not become a straight all at once, but should gradually increase in radius over time (a distance of around {{cvt|40-80|m|ft}} for a line with a maximum speed of about {{cvt|100|km/h|mph}}). Even sharper than curves with no transition are [[reverse curve]]s with no intervening straight track. The [[Cant (road and rail)|superelevation]] must also be transitioned. Higher speeds require longer transitions.

== Vertical curves ==

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 [[normal force|support force]] ''R'' exerted by the track on a train as a function of the curve radius ''r'', the train mass {{mvar|m}}, and the speed {{mvar|v}}, is given by
:<math>R=mg\plusmn\frac{mv^2}{r}</math>
with the second term positive for troughs, negative for crests.  For passenger comfort the ratio of the [[gravitational acceleration]] ''g'' to the [[centripetal acceleration]] ''v<sup>2</sup>/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 trains are sufficiently high-powered that steep slopes are preferable to the reduced speed necessary to navigate horizontal curves around obstacles, or the higher construction costs necessary to tunnel through or bridge over them.  [[High Speed 1]] (section 2) in the UK has a minimum vertical curve radius of {{convert|10000|m|ft|0|abbr=on}}<ref>http://www.whatdotheyknow.com/request/24986/response/79568/attach/3/HS1%20Section%202%20Register%20of%20Infrastructure.pdf - page 19</ref> and [[High Speed 2]], with the higher speed of {{convert|400|km/h|abbr=on}}, stipulates much larger {{convert|56000|m|ft|0|abbr=on}} radii.<ref>http://highspeedrail.dft.gov.uk/sites/highspeedrail.dft.gov.uk/files/hs2-route-engineering.pdf - page 4</ref>  In both these cases the experienced change in weight is less than 7%.

Rail [[well car]]s also risk [[Breakover angle|low clearance]] at the tops of tight crests.

== Problem curves ==
* The [[Australian Standard Garratt]] had [[Driving wheel#Flangeless wheels|flangeless]] leading driving wheels that tended to cause derailments on sharp curves.
* Sharp curves on the [[Port Augusta railway station|Port Augusta]] to [[Hawker, South Australia|Hawker]] line of the [[South Australian Railways]] caused derailment problems when bigger and heavier [[South Australian Railways X class|X class]] locomotives were introduced, requiring realignments to ease the curves.<ref>[[Australian Railway History]] September 2008, p291.</ref>
* {{convert|5|chain|m ft|0|adj=on}} curves on the [[Oberon railway line|Oberon]], [[Tumut and Kunama railway lines|Batlow]], and [[Dorrigo railway line|Dorrigo]] lines, New South Wales limited steam locomotives to the [[0-6-0]] [[New South Wales 19 class locomotive|19 class]].

== List of selected minimum curve radii ==
{{More citations needed section|date=June 2019}}
{| class="wikitable sortable" border="1"
|-
! scope="col" | Radius
! scope="col" class="unsortable" | Location
! scope="col" | Gauge
! scope="col" class="unsortable" | Notes
|-
| {{convert|sortable=on|8000|m|ft|0|abbr=on}} || [[Japan]] || N/A ([[maglev]]) || [[Chūō Shinkansen]] ({{cvt|disp=sqbr|505|km/h}})
|-
| {{convert|sortable=on|7000|m|ft|0|abbr=on}} || rowspan="5" | [[High-speed rail in China|Chinese high&nbsp;speed railway&nbsp;network]] || data-sort-value=1435|{{Track gauge|1435 mm}} || {{cvt|disp=sqbr|350|km/h}}
|-
| {{convert|sortable=on|5500|m|ft|0|abbr=on}} || data-sort-value=1435|{{Track gauge|1435 mm}} || {{cvt|disp=sqbr|250|-|300|km/h}}
|-
| {{convert|sortable=on|4000|m|ft|0|abbr=on}} || data-sort-value=1435|{{Track gauge|1435 mm}} || {{cvt|disp=sqbr|300|km/h}}
|-
| {{convert|sortable=on|3500|m|ft|0|abbr=on}} || data-sort-value=1435|{{Track gauge|1435 mm}} || {{cvt|disp=sqbr|200|-|250|km/h}}
|-
| {{convert|sortable=on|2000|m|ft|0|abbr=on}} || data-sort-value=1435|{{Track gauge|1435 mm}} || {{cvt|disp=sqbr|200|km/h}}
|-
| rowspan="2" | {{convert|sortable=on|1200|m|ft|0|abbr=on}} || rowspan="4" | [[Africa]] || data-sort-value=1435|{{Track gauge|1435 mm}} || Typical of medium-speed railways ({{cvt|disp=sqbr|120|km/h}}) Passenger
|-
| data-sort-value=1435|{{Track gauge|1435 mm}} ||Typical of medium-speed railways ({{cvt|disp=sqbr|80|km/h}}) Freight
|-
| {{convert|sortable=on|800|m|ft|0|abbr=on}} || data-sort-value=1435|{{Track gauge|1435 mm}} || Typical of medium-speed railways ({{cvt|disp=sqbr|120|km/h}}) Passenger
|-
| {{convert|sortable=on|800|m|ft|0|abbr=on}} || data-sort-value=1435|{{Track gauge|1435 mm}} || Typical of medium-speed railways ({{cvt|disp=sqbr|80|km/h}}) Freight
|-
| {{convert|sortable=on|250|m|ft|0|abbr=on}} || [[Democratic Republic of the Congo|DRCongo]] [[Matadi–Kinshasa Railway]] || data-sort-value="1067"|{{Track gauge|1067 mm}} || Deviated {{Track gauge|1067 mm}} line.
|-
| {{convert|sortable=on|240|m|ft|0|abbr=on}} || [[Cougal Spiral (railway)|Border Loop]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || {{convert|5000|LT|t ST|abbr=on|lk=on}} - {{convert|1500|m|ft|0|abbr=on}}
|-
| {{convert|sortable=on|200|m|ft|0|abbr=on}} || [[Wollstonecraft railway station|Wollstonecraft station, Sydney]] || data-sort-value="1435"|{{Track gauge|1435 mm}} 
|-
| {{convert|sortable=on|200|m|ft|0|abbr=on}} || [[Homebush railway station|Homebush]] triangle || data-sort-value="1435"|{{Track gauge|1435 mm}} || {{convert|5000|LT|t ST|abbr=on|lk=on}} - {{convert|1500|m|ft|0|abbr=on}}
|-
| {{convert|sortable=on|190|m|ft|0|abbr=on}} || [[Turkey]]<ref name="Jane" /> || data-sort-value="1435"|{{Track gauge|1435 mm}} || 
|-
| rowspan="2" | {{convert|sortable=on|175|m|ftin|abbr=on|frac=16}} || [[Indian Railways]] || data-sort-value="1676"|{{Track gauge|1676 mm}} || 
|-
| [[Transportation in North America#Railways|North American rail network]]|| data-sort-value="1435"|{{Track gauge|1435 mm}} || Preferred minimum on freight main lines
|-
| {{convert|sortable=on|160|m|ft|0|abbr=on}} || [[Lithgow Zig Zag]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || 40&nbsp;km/h
|-
| {{convert|sortable=on|125|m|ftin|1|abbr=on|frac=16}} || [[Transportation in North America#Railways|North American rail network]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || Minimum radius for general service
|-
| {{convert|sortable=on|120|m|ft|abbr=on}}<ref>{{cite book| author=Paul Garbutt | title=World Metro Systems | chapter=Facts and Figures | publisher=Capital Transport | year=1997 | isbn=1-85414-191-0 | pages=130–131}}</ref> || [[Bay Area Rapid Transit]] || data-sort-value="1676"|{{Track gauge|1676 mm}} ||
|-
| {{convert|sortable=on|100|m|ft|0|abbr=on}} || [[Batlow, New South Wales]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || Rolling stock limited to {{convert|sortable=on|500|LT|t ST|abbr=on|lk=on}} and {{convert|sortable=on|300|m|ft|0|abbr=on}} - restricted to [[New South Wales Z19 class locomotive|NSW Z19 class]] [[0-6-0]] steam locomotives
|-
| {{convert|sortable=on|95|m|ft|0|abbr=on}} || [[Newmarket, New Zealand]] || data-sort-value="1067"|{{Track gauge|1067 mm}} || Extra heavy concrete sleepers<ref>[[Railway Gazette International]] March, 2012, page 23</ref>
|-
| {{convert|sortable=on|87.8|m|ftin|1|abbr=on|frac=16}} || [[North American rail network]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || Absolute minimum radius; not on lines for general service
|-
| {{convert|sortable=on|85|m|ft|0|abbr=on}} || [[Windberg Railway]] ([[:de:Windbergbahn]]) || data-sort-value="1435"|{{Track gauge|1435 mm}} || (between [[Freital]]-Birkigt and [[Dresden]]-Gittersee) - restrictions to wheelbase
|-
| {{convert|sortable=on|80|m|ft|0|abbr=on}} || [[Queensland Rail]]ways || data-sort-value="1067"|{{Track gauge|1067 mm}} || Central Line between Bogantungan and Hannam's Gap
|-
| {{convert|sortable=on|70|m|ft|abbr=on}} || [[JFK Airtrain]]|| data-sort-value="1435"|{{Track gauge|1435 mm}} ||
|-
| {{convert|sortable=on|68.6|m|abbr=on|ftin|frac=16}} || [[Washington Metro]]<ref>{{cite web|url=http://www.wmata.com/business/procurement_and_contracting/solicitations/uploads/RFP%20Energy%20Storage-Attachment%20B-Rail%20Car%20Performance%20for%20Design-Simulation.pdf|title=WMATA Summary – Level Rail Car Performance For Design And Simulation|date=2013-10-13|publisher=WMATA|author=<!--Staff writer(s); no by-line.-->|access-date=October 15, 2014|url-status=dead|archive-url=https://web.archive.org/web/20160114052554/http://www.wmata.com/business/procurement_and_contracting/solicitations/uploads/RFP%20Energy%20Storage-Attachment%20B-Rail%20Car%20Performance%20for%20Design-Simulation.pdf|archive-date=January 14, 2016}}</ref>|| data-sort-value="1429"|{{Track gauge|4ft8.25in}} ||
|-
| {{convert|sortable=on|61|m|ft|0|abbr=on}} || [[London Underground]] [[Central line (London Underground)|Central line]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || (between White City and Shepherd's Bush)
|-
|{{convert|sortable=on|53|m|ft|0|abbr=on}}
|[[New York City Subway]]
|{{Track gauge|1435 mm}}
|Several 175-foot curves of the BMT Eastern Division do not permit the 75-foot cars through them<ref name="DCP-Broadway-Junction-Pt3">{{cite web |date=2007 |title=3.3.3 A, C (Fulton Street) Line Services and Structural Issues |url=https://www1.nyc.gov/html/dcp/pdf/transportation/broadway_junction_pt3.pdf |url-status=dead |archive-url=https://web.archive.org/web/20100606141009/http://www.nyc.gov/html/dcp/pdf/transportation/broadway_junction_pt3.pdf |archive-date=June 6, 2010 |access-date=February 9, 2016 |publisher=New York City Department of City Planning |location=New York}}</ref>
|-
| rowspan="3" | {{convert|sortable=on|50|m|ft|abbr=on}} || [[Gotham Curve]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || [[Cromford and High Peak Railway]], [[Derbyshire]], England until 1967
|-
| [[Matadi-Kinshasa Railway]]|| data-sort-value="762"|{{Track gauge|762 mm}} || original {{Track gauge|762 mm}} line.
|-
| [[Welsh Highland Railway]] || data-sort-value="600"|{{Track gauge|600 mm}} ||
|-
| {{convert|sortable=on|45|m|ft|0|abbr=on}} || [[Bernina Railway]] || data-sort-value="1000"|{{Track gauge|1000 mm}} ||
|-
| rowspan="2" | {{convert|sortable=on|40|m|ft|0|abbr=on}} || [[Welsh Highland Railway]] || data-sort-value="600"|{{Track gauge|600 mm}} || on original line at [[Beddgelert]]
|-
| [[Narrow-gauge lines of the Victorian Railways|Victorian Narrow Gauge]]|| data-sort-value="762"|{{Track gauge|762 mm}} || {{convert|sortable=on|16|km/h|mph|0|abbr=on|disp=or}} on&nbsp;curves ({{convert|sortable=on|32|km/h|mph|0|abbr=on|disp=or}} on&nbsp;straightaways)
|-
| {{convert|sortable=on|37.47|m|ftin|abbr=on|disp=or|frac=16}} (48°) || [[Kalka-Shimla Railway]] || data-sort-value="762"|{{Track gauge|762 mm}} || 
|-
| {{convert|sortable=on|30|m|ft|abbr=on|0}} || [[Metromover]] || N/A (monorail) || Rubber-tired, monorail-guided light rail downtown [[people mover]] system.<ref name="MMexpstudyexecutive">{{cite web |url=http://miamidadempo.org/library/studies/metromover-system-expansion-study-executive-summary-2014-09.pdf |title=Metromover System Expansion Study |publisher=Miami-Dade MPO |date=September 2014 |access-date=February 13, 2015 |archive-url=https://web.archive.org/web/20150214005617/http://miamidadempo.org/library/studies/metromover-system-expansion-study-executive-summary-2014-09.pdf |archive-date=February 14, 2015 |url-status=dead }}</ref>
|- 
| {{convert|sortable=on|29|m|ft|abbr=on}} || [[New York City Subway]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || <ref>[[Railway Gazette International]], July 2012, p18</ref>
|-
| {{convert|sortable=on|27|m|ft|0|abbr=on}} || [[Chicago 'L']]|| data-sort-value="1435"|{{Track gauge|1435 mm}} ||
|-
| rowspan="2" | {{convert|sortable=on|25|m|ft|0|abbr=on}} || [[Sydney Steam Motor Tram]] <br/> 0-4-0 || rowspan="2" data-sort-value="1435" |{{Track gauge|1435 mm}} || Hauling 3 trailers 
|-
|[[Luas]]
|<ref>{{Cite web |title=Luas Track and Power Supply |url=https://www.tii.ie/en/public-transport/luas/track-power/ |access-date=2024-11-24 |website=www.tii.ie}}</ref>
|-
| {{convert|sortable=on|22|m|ft|0|abbr=on}} || [[Warsaw Commuter Railway]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || Depot tracks in Grodzisk Mazowiecki, Poland<ref>{{Cite web | url=https://www.wkd.com.pl/o-wkd/charakterystyka-linii.html |title=Charakterystyka linii |trans-title=Line parameters | website=WKD |lang=pl }}</ref>
|-
| {{convert|sortable=on|21.2|m|ftin|abbr=on|frac=16}} || [[Darjeeling Himalayan Railway]] || data-sort-value="610"|{{Track gauge|610 mm}} || Sharpest curves were originally {{convert|13.7|m|ftin|0|abbr=on|frac=16}}<ref>Trains: The Early Years, page 51, H. F. Ullmann,Getty Images, {{ISBN|978-3833-16183-4}}</ref>
|-
| {{convert|sortable=on|18.25|m|ftin|abbr=on|frac=16}} || [[Matheran Hill Railway]] || data-sort-value="610"|{{Track gauge|610 mm}} || 1 in 20 (5%); {{convert|8|km/h|mph|0|abbr=on|disp=or}} on curve; {{convert|20|km/h|mph|0|abbr=on|disp=or}} on straight
|-
| {{convert|sortable=on|15.24|m|ftin|2|abbr=on|frac=16}} || rowspan="2" | [[Streetcars in New Orleans]]<ref>[http://www.lightrailnow.org/features/f_lrt_2007-02a.htm Lightrail now] New Orleans RTA/Brookville streetcar</ref> || data-sort-value="1587.5"|{{Track gauge|5 ft 2 1/2 in|first=met}} || Revenue service
|-
| {{convert|sortable=on|8.53|m|ftin|2|abbr=on|frac=16}} || data-sort-value="1587.5"|{{Track gauge|5 ft 2 1/2 in|first=met}} || Yard tracks
|-
| {{convert|sortable=on|13.11|m|ftin|2|abbr=on|frac=16}} || [[San Francisco Municipal Railway]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || Light rail, former streetcar system
|-
| {{convert|sortable=on|10.973|m|ftin|0|abbr=on|frac=16}} || [[Toronto Streetcar System]] || data-sort-value="1495"|{{Track gauge|toronto|first=met}} ||
|-
| {{convert|sortable=on|10.67|m|ftin|0|abbr=on|frac=16}} || [[Taunton Tramway]] || data-sort-value="1067"|{{Track gauge|1067 mm}} ||
|-
| {{convert|sortable=on|10.058|m|ftin|3|abbr=on|frac=16}} || [[Green Line (MBTA)|Boston Green Line]] || data-sort-value="1435"|{{Track gauge|1435 mm}} ||
|-
| {{convert|sortable=on|10.06|m|ftin|3|abbr=on|frac=16}} || [[Newark Light Rail]] || data-sort-value="1435"|{{Track gauge|1435 mm}} || 
|-
| {{convert|sortable=on|4.9|m|ftin|abbr=on|frac=16}} || [[Chicago Tunnel Company]] || data-sort-value="610"|{{Track gauge|610 mm}} || {{convert|sortable=on|6.1|m|ftin|abbr=on|frac=16}} in [[grand union]]s. Not in use.
|}

== See also ==
{{div col|colwidth=22em}}
* [[Breakover angle]]
* [[:Category:Articulated locomotives]]
* [[Degree of curvature#Formula from chord length|Degree of curvature]], [[civil engineering]] 
* [[Lateral motion device]]
* [[Longest trains]]
* [[Matheran Hill Railway]]
* [[Radius]]
* [[Radius of curvature (applications)]]<ref>[http://openrails.org/files/superelevation_v1.pdf Superevevation]</ref>
* [[Railway systems engineering]]
* [[Track transition curve]]
* [[Turning radius]]

{{div col end}}

== References ==
{{reflist}}

== External links ==
* {{cite book| last1 = Hilton| first1 = George W.| last2 = Due| first2 = John Fitzgerald| title = The Electric Interurban Railways in America| url = https://books.google.com/books?id=J2zH-zcuU-MC&pg=PA51| access-date = 10 June 2014| date = 1 January 2000| publisher = Stanford University Press| isbn = 978-0-8047-4014-2}}

{{Rail tracks}}
{{Railway track layouts}}

[[Category:Track geometry]]
[[Category:Radii]]