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In this paper, we study the shape of bifurcation curve $S_{L}$ of positive solutions for the Minkowski-curvature problem \begin{equation*} \begin{cases} -\left( \dfrac{u^{\prime }(x)}{\sqrt{1-\left( {u^{\prime }(x)}\right) ^{2}}} \right) ^{\prime }=\lambda \left( -\varepsilon u^{3}+u^{2}+u+1\right) ,& -L<x<L, \\ u(-L)=u(L)=0, \end{cases} \end{equation*} where $\lambda ,\varepsilon >0$ are bifurcation parameters and $L>0$ is an evolution parameter. We prove that there exists $\varepsilon _{0}>0$ such that the bifurcation curve $S_{L}$ is monotone increasing for all $L>0$ if $ \varepsilon \geq \varepsilon _{0}$, and the bifurcation curve $S_{L}$ is from monotone increasing to S-shaped for varying $L>0$ if $0<\varepsilon <\varepsilon _{0}.$