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We show that the right (resp. left) $q$-deformed rational numbers associated to $\\frac{r}{s}$, in the sense of Morier-Genoud-Ovsienko (resp. Bapat-Becker-Licata) can be naturally calculated by the $\\mathfrak{q}$-intersection between $\\widehat{\\eta}_{\\frac{r}{s}}$ and $\\mathbf{A}$ (resp. dual arc system $\\mathbf{A}^*$). 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We associate a bigraded simple closed arc $\\widehat{\\eta}_{\\frac{r}{s}}$ on $\\mathbf{D}_{3}$ to any rational number $\\frac{r}{s}\\in\\overline{\\mathbb{Q}}=\\mathbb{Q}\\cup\\{\\infty\\}$. We show that the right (resp. left) $q$-deformed rational numbers associated to $\\frac{r}{s}$, in the sense of Morier-Genoud-Ovsienko (resp. Bapat-Becker-Licata) can be naturally calculated by the $\\mathfrak{q}$-intersection between $\\widehat{\\eta}_{\\frac{r}{s}}$ and $\\mathbf{A}$ (resp. dual arc system $\\mathbf{A}^*$). 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