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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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    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.2012-02-13
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    It seems this is explained rather well at [Wikipedia](http://en.wikipedia.org/wiki/Gimbal_lock), including visualizations. It would be easier to help you if you pointed out specifically what part of that article you don't understand.2012-02-13
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    Related: http://math.stackexchange.com/questions/8980/euler-angles-and-gimbal-lock2012-02-13
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    @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 $f$.2012-02-13

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