Editing Gimbal lock (section)

===In two dimensions===
[[File:Theodolite vermeer.svg|left|thumb]]
Gimbal lock can occur in gimbal systems with two degrees of freedom such as a [[theodolite]] with rotations about an [[azimuth]] (horizontal angle) and elevation (vertical angle). These two-dimensional systems can gimbal lock at [[zenith]] and [[nadir]], because at those points azimuth is not well-defined, and rotation in the azimuth direction does not change the direction the theodolite is pointing.

Consider tracking a helicopter flying towards the theodolite from the horizon. The theodolite is a telescope mounted on a tripod so that it can move in azimuth and elevation to track the helicopter. The helicopter flies towards the theodolite and is tracked by the telescope in elevation and azimuth. The helicopter flies immediately above the tripod (i.e. it is at zenith) when it changes direction and flies at 90 degrees to its previous course. The telescope cannot track this maneuver without a discontinuous jump in one or both of the gimbal orientations. There is no continuous motion that allows it to follow the target. It is in gimbal lock. So there is an infinity of directions around zenith for which the telescope cannot continuously track all movements of a target.<ref>{{cite web|url= http://www.madsci.org/posts/archives/aug98/896993617.Eg.r.html |title= Re: What is meant by the term gimbal lock? |author= Adrian Popa |date= June 4, 1998}}</ref> Note that even if the helicopter does not pass through zenith, but only ''near'' zenith, so that gimbal lock does not occur, the system must still move exceptionally rapidly to track it, as it rapidly passes from one bearing to the other. The closer to zenith the nearest point is, the faster this must be done, and if it actually goes through zenith, the limit of these "increasingly rapid" movements becomes ''infinitely'' fast, namely discontinuous.

To recover from gimbal lock the user has to go around the zenith – explicitly: reduce the elevation, change the azimuth to match the azimuth of the target, then change the elevation to match the target.

Mathematically, this corresponds to the fact that [[spherical coordinates]] do not define a [[coordinate chart]] on the sphere at zenith and nadir. Alternatively, the corresponding map ''T''<sup>2</sup>&rarr;''S''<sup>2</sup> from the [[torus]] ''T''<sup>2</sup> to the sphere ''S''<sup>2</sup> (given by the point with given azimuth and elevation) is not a [[covering map]] at these points.