# Why does aliasing cause loss of a degree of freedom in Euler angles?

I'm reading a book on 3D game math where the author points out that when using Euler angles the same orientation can be reached by doing two different operations; say rotating a cube 90 degrees around Y-axis is the same as rotating 90 degrees around X, Y and Z-axis successively.

While I have experimentally verified that aliasing is persent when using Euler angles to represent orientation (since the mapping is many-to-one), I'm unable to understand why this singularity leads to the loss of a degree of freedom (Gimbal Lock).

Can someone mathematically explain the reason? I see no loss here, I'll still be able to orient the cube whichever way I want it to be oriented, then where's the lock?

Many people pointed me to videos in YouTube, which explains gimbal lock using 3D animations where a gimbal gets aligned to another and rotation is arrested; while this is a visual display of gimbal lock, I don't understand why something which is a mechanical/physical limitation also applies to the mathematical model of assoicating orientation with 3 (Euler) angles.

• math.stackexchange.com/questions/8980/… seems to be a closely related question, but the difference is that it never talks about aliasing and again it uses a mechanical model to explain a mathematical issue inherent in a system. – legends2k Mar 5 '13 at 17:33
• I'm surprised that no one has commented/answered with the link between singularity and homeomorphism. – legends2k Mar 28 '13 at 12:39
• This answer nicely puts it: "Euler Angles" you can think of as a function (S1)3→SO3 or R3→SO3. The derivative of this function does not always have rank 3, so you have degenerate submanifolds where the function is many-to-one. In this special case that's called "gimbal lock". Since in Euler angles multiple values represent the same orientation (aliasing) the mapping isn't one to one, so we've no way to distinguish/record the variation in one degree since the rank becomes less then 3. – legends2k May 9 '14 at 14:05

## 1 Answer

I'll still be able to orient the cube whichever way I want it to be oriented, then where's the lock?

I believe that you are correct as long as you look at things statically.

Problems start in physical models when you start moving things around. Probably the closest thing to such a movement in the mathematical world is an infinitesimal change in orientation. Think about applications from calculus, e.g. differentiating something with respect to changes in orientation. In that case you'd conceptually perform infinitesimally small changes to your parameters to describe infinitesimally small changes to the rotation itself.

Except for the singular gimbal-locked situations, where some infinitesimal changes to the parameters don't affect the rotation at all, and on the other hand some infinitesimal changes to the rotation would require a finite (i.e. not infinitesimally small) change to your parameters. The latter breaks a lot of mathematical machinery.