Gimbal lock and zero jacobian determinant

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 $f(p+\epsilon)$?

• 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. – Jyrki Lahtonen Feb 13 '12 at 9:09
• 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. – joriki Feb 13 '12 at 10:14
• – Hans Lundmark Feb 13 '12 at 11:37
• @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 $f$. – user782220 Feb 13 '12 at 22:43