Track transition curve

In the UK, only from 1845, when legislation and land costs began to constrain the laying out of rail routes and tighter curves were necessary, were the principles beginning to be applied in practice.The 'true spiral', whose curvature is exactly linear in arclength, requires more sophisticated mathematics (in particular, the ability to integrate its intrinsic equation) to compute than the proposals that were cited by Rankine.The degree of banking in railroad track is typically expressed as the difference in elevation of the two rails, commonly quantified and referred to as the superelevation.Such difference in the elevation of the rails is intended to compensate for the centripetal acceleration needed for an object to move along a curved path, so that the lateral acceleration experienced by passengers/the cargo load will be minimized, which enhances passenger comfort/reduces the chance of load shifting (movement of cargo during transit, causing accidents and damage).The simplest and most commonly used form of transition curve is that in which the superelevation and horizontal curvature both vary linearly with distance along the track.
The red Euler spiral is an example of an easement curve between a blue straight line and a circular arc, shown in green.
This sign aside a railroad (between Ghent and Bruges ) indicates the start of the transition curve. A parabolic curve ( POB ) is used.
Brusio spiral viaduct and railway (Switzerland, built 1908), from above
Euler spiralBrugesparabolic curverailroad tracktangentslateral (centripetal) accelerationearly railroadsRankine'scurve of sinesWilliam GravattWilliam Froudeelastic curveBrusio spiral viaductintrinsic equationLeonhard EulerArthur N. Talbotclothoidtrack geometrythree-dimensionalquadraticallytangentcircular arcsuperelevationcentripetal accelerationCartesian coordinatesFresnel integralsDegree of curvatureMinimum railway curve radiusRailway systems engineeringRankine, WilliamArchibald, Raymond ClareAmerican Railway Engineering and Maintenance-of-Way AssociationTracks(history)Axe tiesBallastBaulk roadBreather switchClip and scotchDate nailFastening systemFishplateLadder trackMinimum radiusProfileTie/SleeperBalloon loopClassification yardHeadshuntPocket trackJunctionGauntlet trackGuide barPassing loopTrack gaugedual gaugeRail tracktramway trackRail yardRailway electrificationoverhead linesthird railground-level power supplyRailway turntableTransfer table (traverser)Roll waySidingrefuge sidingSwitchWater craneWater troughSignallingAnti-trespass panelsBlock postBuffer stopCatch pointsDefect detectorDerailerGuard railInterlockingLevel crossingLoading gaugePlatform screen doorsRailway signalSignalling controlStructure gaugeSignal bridgeTell-taleTrain stopWayside hornCoaling towerMotive power depotRailway workshopPlatformRoundhousefor trainsfor goodsStationbuildingWater stopIndustrialMilitaryPrivate