# Train track

﻿
Train track

In the mathematical area of topology, a train track is a family of curves embedded on a surface, meeting the following conditions:
#The curves meet at a finite set of vertices called "switches".
#Away from the switches, the curves are smooth and do not touch each other.
#At each switch, three curves meet with the same tangent line, with two curves entering from one direction and one from the other.

The main application of train tracks in mathematics is to study laminations of surfaces, that is, partitions of closed subsets of surfaces into unions of smooth curves. Train tracks have also been used in graph drawing.

Train tracks and laminations

A lamination of a surface is a partition of a closed subset of the surface into smooth curves. The study of train tracks was originally motivated by the following observation: If a generic lamination on a surface is looked at from a distance by a myopic person, it will look like a train track.

A switch in a train track models a point where two families of parallel curves in the lamination merge together to become a single family, as shown in the illustration. Note that, although the switch consists of three curves ending in and intersecting at a single point, the curves in the lamination do not have endpoints and do not intersect each other.

For this application of train tracks to laminations, it is often important to constrain the shapes that can be formed by connected components of the surface between the curves of the track. For instance, Penner and Harer require that each such component, when glued to a copy of itself along its boundary to form a smooth surface with cusps, have negative cusped Euler characteristic.

A train track with weights, or weighted train track or measured train track, consists of a train track with a non-negative real number, called a "weight", assigned to each branch. The weights can be used to model which of the curves in a parallel family of curves from a lamination are split to which sides of the switch. Weights must satisfy the following switch condition: The weight assigned to the ingoing branch at a switch should equal the sum of the weights assigned to the branches outgoing from that switch.Weights are closely related to the notion of carrying. A train track is said to carry a lamination if there is a train track neighborhood such that every leaf of the lamination is contained in the neighborhood and intersects each vertical fiber transversely. If each vertical fiber has nontrivial intersection with some leaf, then the lamination is fully carried by the train track.

References

*cite book
author = Penner, R. C., with Harer, J. L.
title = Combinatorics of Train Tracks
publisher = Princeton University Press, Annals of Mathematics Studies
year = 1992
isbn = 0-691-02531-2

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• train track — noun a) railroad track b) A set of curves lying in a surface, meeting one another at their respective endpoints in a specific way that resembles railroad tracks …   Wiktionary

• Track (rail transport) — Permanent way redirects here. For other uses, see Permanent way (disambiguation). Twin rail tracks in a wooded area …   Wikipedia

• track — track1 [ træk ] noun *** ▸ 1 rough path/road ▸ 2 what train moves along ▸ 3 running/racing course ▸ 4 part on which something moves ▸ 5 direction or way ▸ 6 mark ▸ 7 recorded piece of music ▸ + PHRASES 1. ) count a path or road with a rough… …   Usage of the words and phrases in modern English

• track — 1. noun /træk/ a) A mark left by something that has passed along; as, the track, or wake, of a ship; the track of a meteor; the track of a sled or a wheel. Im going to try out for track next week. b) A mark or impression left by the foot, either… …   Wiktionary

• Train order operation — is a system by which the railroads of North America conveyed operating instructions before the days of centralized traffic control, direct traffic control, and the use of track warrants conveyed by radio.Train order operation still exists on a… …   Wikipedia

• Track warrant — Track warrants are systematized permissions used on some rail lines to authorize use of the main line instead of signals. Train crews receive track warrants by radio, phone, or electronic transmission from a dispatcher. Track warrants are issued… …   Wikipedia

• Train shunting puzzle — Train shunting puzzles, also often called railway shunting puzzles or railroad switching puzzles, are a type of transport puzzle.Shunting puzzles usually consist of a specific track layout, a set of initial conditions (typically the starting… …   Wikipedia

• Train speed optimization — Train speed optimization, also known as Zuglaufoptimierung, is a system that reduces the need for trains to brake and accelerate, resulting in smoother and more efficient operation.While train speed optimization needs some technical… …   Wikipedia

• Train routes in the Netherlands — Train routes by timetable numberTrain routes in the Netherlands are identified by numbers. The routes are listed here using the official abbreviations for the station names, see list of stations, with their official abbreviations, and in the… …   Wikipedia

• Train of Thought (Dream Theater album) — Train of Thought Cover art by Jerry Uelsmann Studio album by Dream Theater …   Wikipedia

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.