I know in a compact Riemannian manifold, this result on equivalence of metrics hold. As a consequence, we can take an arbitrary compact Riemannian $N$-manifold $(M,g)$ with its covering by the charts $(U_k,\phi_k)$ for $k=1,2,\dots,m$ and we have the following relation $$\frac1Q|\phi_k^{-1}(x)-\phi_k^{-1}(y)|\leq d_g(x,y)\leq Q|\phi_k^{-1}(x)-\phi_k^{-1}(y)|$$ for all $x,y\in M$ and some $Q>1$. As usual, $d_g$ is the geodesic distance.
I want to know if there is a way to explicitly calculate this constant $Q$. If not, is it possible to say upto what extent this constant $Q$ could depend on the manifold $(M,g)$? Any help is appreciated.