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I'm having a hard time visualizing the gimbal lock problem. Suppose $p=(a,b,c)$ is a point where the euler angle $f:\mathbb R^3\to SO_3$ has zero jacobian determinant. Then to say that $p$ is where gimbal lock occurs can be mathematically formulated as saying that $f$ has no bijection from a small area around $p$ to a small ball around $f(p)$.

So there are rotations very close to $f(p)$ which cannot be represented as $f(p+\epsilon)$ for small $\epsilon$. I am having a hard time visualizing this. Does someone have a way to see that such rotations cannot be represented as $f(p+\epsilon)$?

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  • $\begingroup$ The gimbal locks that I can imagine in my head usually mean that the vector $p$ has such a value that a small change in the first component has the same effect on the frame as a small change in the third component. I'm not positive that this implies local non-bijectivity, but it does imply that the inverse mapping is not differentiable at that point. $\endgroup$ – Jyrki Lahtonen Feb 13 '12 at 9:09
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    $\begingroup$ It seems this is explained rather well at Wikipedia, including visualizations. It would be easier to help you if you pointed out specifically what part of that article you don't understand. $\endgroup$ – joriki Feb 13 '12 at 10:14
  • $\begingroup$ Related: math.stackexchange.com/questions/8980/… $\endgroup$ – Hans Lundmark Feb 13 '12 at 11:37
  • $\begingroup$ @joriki I don't think wikipedia gives a full explanation. The image of a small ball around $p$ under $f$ does not contain a small ball around $f(p)$. A full explanation would give an explicit description of the geometry of the image of the ball around $p$ under $f$. $\endgroup$ – user782220 Feb 13 '12 at 22:43

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