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# Part 3: Gimbal Lock

Modified 2020-10-22 by sageshoyu

The orientation of an object in 3D space can be described by a set of three values: , where is roll, is pitch, and is yaw.

Mathematically, any point on an object that undergoes rotation will have a new coordinate calculated as follows: Where:

Ideally, we would hope that the parameters are enough to rotate any point (distance from the origin) to any other point (also distance from the origin, since rotations do not change distance). Upon closer thought, it would seem as if we have more than enough parameters to do this, since it only takes two parameters to describe all points on the 3D unit sphere

However, this intuition is a bit off. If any one parameter is held fixed, it may be impossible for to be rotated to some other by varying the remaining two parameters. Moreover, if a certain parameter is set to a certain problematic value, then varying the remaining two parameters will either sweep out a circle (not a sphere!), or not affect at all, depending on what is. This result is way different from what we expected! The name for this degenerate case is gimbal lock.

## Questions

Modified 2020-09-08 by sageshoyu

1. Suppose an airplane increases its pitch to (i.e. ):

Let denote the rotation matrix for . Prove that

2. Consider the point on the pitched airplane, i.e. the tip of the wing. Does there exist any such that:

For ?