Let 6 curves $(\gamma_i(.))_{0<i<7}$ such that $\gamma_i =(p_{e_x},p_{e_y},p_{h_x},p_{h_y},p_P)$ and represented in the graph below.
Does these curves look homothetic ? that is one can deduce $\gamma_k$ from $\gamma_j$ by an homothetie. I am looking for necessary conditions, because for instance in the graph for $p_P$, the curves are intersecting not in 0, is it possible ?
Let $\gamma_1$ and $\gamma_2$ two curves.
Are there some constants k and b such that, for all t $$\gamma_1(t) = k \gamma_2(t) +b\; ? $$
We have $$p_{e_y}=0=p_{h_y}$$
So if $\gamma_1$ and $\gamma_2$ are homothetic, then b=0 which is not possible since in the Pp-graph, the curves are not intersecting in $0$... so there are not homothetic