In a projective plane (i.e. two-dimensional) $\mathbb P$, we call a general homology a projective transformation $h:\mathbb P\to\mathbb P$ such that $h$ has a line of fixed points $L$ called the axis of $h$, a fixed point $O\not\in L$ called the center of $h$ and has the property that any line through $O$ is invariant under $h$. How should one study the possible situations that can arise with the composition $h\circ g$ of two general homologies with the same center $O$, and different axes $L$ and $R$.
Should I try to analyze this geometrically? (since it's the plane, certain representative drawings are possible, which are then backed by algebraic considerations), or should I use pure algebra and consider different Jordan form, bases, etc. configurations? I know the question is general so any single case study would be a great answer.