Editing Covering space (section)

== Applications ==
[[Image:Rotating gimbal-xyz.gif|thumb|300px|[[Gimbal lock]] occurs because any map {{nowrap|''T''<sup>3</sup> → '''RP'''<sup>3</sup>}} is not a covering map. In particular, the relevant map carries any element of ''T''<sup>3</sup>, that is, an ordered triple (a,b,c) of angles (real numbers mod&nbsp;2{{pi}}), to the composition of the three coordinate axis rotations R<sub>x</sub>(a)<math>\circ</math>R<sub>y</sub>(b)<math>\circ</math>R<sub>z</sub>(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group SO(3), which is topologically '''RP'''<sup>3</sup>.

This animation shows a set of three gimbals mounted together to allow ''three'' degrees of freedom. When all three gimbals are lined up (in the same plane), the system can only move in two dimensions from this configuration, not three, and is in ''gimbal lock''. In this case it can pitch or yaw, but not roll (rotate in the plane that the axes all lie in).]]

An important practical application of covering spaces occurs in [[charts on SO(3)]], the [[rotation group SO(3)|rotation group]]. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in [[navigation]], [[nautical engineering]], and [[aerospace engineering]], among many other uses. Topologically, SO(3) is the [[real projective space]] '''RP'''<sup>3</sup>, with fundamental group '''Z'''/2, and only (non-trivial) covering space the hypersphere ''S''<sup>3</sup>, which is the group [[spin group|Spin(3)]], and represented by the unit [[quaternions]]. Thus quaternions are a preferred method for representing spatial rotations – see [[quaternions and spatial rotation]].

However, it is often desirable to represent rotations by a set of three numbers, known as [[Euler angles]] (in numerous variants), both because this is conceptually simpler for someone familiar with planar rotation, and because one can build a combination of three [[gimbal]]s to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus ''T''<sup>3</sup> of three angles to the real projective space '''RP'''<sup>3</sup> of rotations, and the resulting map has imperfections due to this map being unable to be a covering map. Specifically, the failure of the map to be a local homeomorphism at certain points is referred to as [[gimbal lock]], and is demonstrated in the animation at the right – at some points (when the axes are coplanar) the [[Rank (differential topology)|rank]] of the map is 2, rather than 3, meaning that only 2 dimensions of rotations can be realized from that point by changing the angles. This causes problems in applications, and is formalized by the notion of a covering space.