This might be a trivial question; however, it got me quite confused. Let's say I have a 2D plane, and I am randomly put ellipses in this plane. Then if I don't care about the positions and rotations of these ellipses in the plane, I can classify all of them with two numbers (a, b) which, e.g. correspond to minor and major axes. I can organise these two numbers in a matrix
$$ M_{ij} = \begin{pmatrix} 1/a^2 & 0\\ 0 & 1/b^2 \end{pmatrix}, $$ so that now any of these ellipses, positioned anywhere in the plane, however rotated are described by the expression:
$$ (r - r_0)_n(r - r_0)_m R_{ni}(\phi)R_{mj}(\phi)M_{ij}=1,$$
where $R_{nm}(\phi)$ is a rotation matrix in the plane
$$ R_{ij}(\phi) = \begin{pmatrix} \cos \phi & - \sin \phi\\ \sin \phi & \cos \phi \end{pmatrix} $$
that rotates the ellipse around the origin by angle $\phi$, and $r_0=(x_0,y_0)$ is a vector that displaces the ellipse to some point at position $(x_0,y_0)$ away from the origin. My point here is simply that I am able to classify all the ellipses in the plane by two parameters (a,b) and the rest are just rotations and displacements.
Now, my question is how to do the same type of descriptions for triangles? I naturally expect I should be able to use just three numbers (e.g. lengths of three sides) to classify all the triangles. And the rest should be again just displacements and rotations the same as above. But how can I do that? Can one write down some equations analogous to what I have above for ellipse that would describe this procedure of rotating and displacing the triangle?