Relationship between the $\mathtt{Exp}^{-1}$ mapping of three points on a common geodesic on a Riemann manifold

Question

Suppose that there is a Riemann manifold $\mathcal{M}$ and $x,z$ be two point on $\mathcal{M}$. Let \begin{equation} y = \mathtt{Exp}_{x}\left( t \cdot \mathtt{Exp}_{x}^{-1} \left( z \right)\right) \end{equation} Thus $y$ lies on the minimizing geodesic between $x$ and $z$. Consider three vectors $\mathtt{Exp}_{v}^{-1}(x),\mathtt{Exp}_{v}^{-1}(y),\mathtt{Exp}_{v}^{-1}(z)$, they all live in the tangent space of $\mathcal{M}$ at $v$. If $v$ does not lie on the geodesic between $x$ and $z$, then $\mathtt{Exp}_{v}^{-1}(x)$ and $\mathtt{Exp}_{v}^{-1}(z)$ are linearly independent, and thus there exists $t_1$ and $t_2$ such that \begin{equation} \mathtt{Exp}_{v}^{-1}(y) = t_1 \cdot \mathtt{Exp}_{v}^{-1}(x) + t_2 \cdot \mathtt{Exp}_{v}^{-1}(z) \end{equation} My question is, given $t$ and some curvature assumptions of $\mathcal{M}$, is it possible to obtain some bounds of $t_1$ and $t_2$ (ideally independent of $v$)?