# 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