I have a planar object, say a polygon $A_1A_2\ldots A_n$ in the 3-dimensional Euclidean space. It is translated by a vector $v$ and rotated by a rotation matrix $R$, and the resulting image is $A_1'A_2'\ldots A_n'$. So $$A_i'=R(A_i+v).$$ If the coordinates of all $A_i$ and $A_i'$ are known, what is the best way to solve for $R$ and $v$? The objective is to get a simple formula/algorithm and minimize the number $n$ of points needed, assuming the points are "independent" in some reasonable sense.
One possible solution is to write out $R$ as a matrix of 9 variables and $v$ as a vector of 3 variables, so that we get a linear system of $n$ equations. Then $n=12$ would be enough to solve for the 12 variables (assuming the equations are "independent").
I suspect $n$ can be as small as 3. In other words:
Given a triangle $ABC$ and its congruent image $A'B'C'$. Is it possible to find a rotation and a translation that maps $A$, $B$ and $C$ to $A'$, $B'$ and $C'$, respectively? (Under reasonable assumption, i.e., $ABC$ is not degenerate, not collinear, etc.)