For example, an airplane uses three references, pitch (angle up/down), yaw (angle left/right on a horizontal axis) and roll (angle left/right on the vertical axis). If an airplane heads straight up or down (change of pitch), one other reference (the yaw) is cancelled, one loses a dimension of rotation, because there is always a value for one angle of rotation that yields infinite values of the other two angles (in this case, the yaw).

Compare a similar problem with tangents on triangles -- say one has a right triangle ABC, with angle ACB=90. Consider angle BAC. tan(BAC) = BC/AC, but BAC is less than 90 degrees. We can make angle BAC closer and closer to 90 degrees by increasing the length of BC, but as we keep doing this, AC stays the same so the ratio BC/AC gets infinitely large. So tan(90) has no geometric meaning. One leg of the triangle become infinitely long and never meets the other leg.

Another real world comparison is latitude and longitude. At the poles, (latitude 90 degrees north or south), the definition of longitude becomes meaningless (as all longitude lines meet in a singularity).

See also:

- Yaw, pitch and roll
- Charts on SO(3)