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.
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.
