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I am not an expert in this so I hope this doesn't sound so stupid: what is the common metric used when studying the Euclidean Group $\mathrm{E}(3)$. One could probably ask the same thing for (homogenous) transformation group $\mathrm{SE}(3)$. For the latter I have seen use of Riemannian metric. For both I am guessing (as this is not my field) that there are some kind of manifold structure involved, so I can't help to think that there are also good metrics (by good I mean metric that is considered "practical" when one considers rigid body motions).

Furthermore, is there a way to visualize the manifolds from $\mathrm{E}(3)$ and $\mathrm{SE}(3)$? Any good reference in this area?

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    $\begingroup$ Just for clarification, do you mean metric in the sense of a distance function or metric in the sense of a Riemannian metric? $\endgroup$ – Willie Wong Nov 25 '11 at 13:53
  • $\begingroup$ I meant metric in the sense of a distance function. $\endgroup$ – Jose Nov 25 '11 at 14:42
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There are several natural metrics that you can use on transformation groups. For example, for any topological group you could ask for a left or a right or a bi-invariant metric. Another natural way to define a metric is via how it acts on a space. If you really want to think about the isometry group of $\mathbb R^3$, I suppose this is the direction you'd want to go. So let $f$ and $g$ be isometries of $\mathbb R^3$. Define

$$d(f,g) = \max \{ |f(x)-g(x)| : x \in \mathbb R^3, |x| \leq 1 \}$$

This has the advantage that it's left-invariant already. IMO this is a pretty good metric. If $f$ and $g$ fix the origin they're linear and this is the norm of $f-g$, which is considered a fairly standard metric in linear algebra.

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