Suppose, for each pair $(\lambda,\nu)\in\operatorname{SC}(S)\times\operatorname{SC}(T)$ we have a bi-polyhedral Kähler affine structure on ${\Sigma\backslash D}$ of type $(\lambda,\nu)$, which varies continuously with $(\lambda,\nu)$ in the Hausdorff topology of metric structures on $\Sigma$. We call such a family projective if:

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  For any linear functions $l\in\mathbb{R}^{d},\ l^{\vee}\in(\mathbb{R}^{d})^{*}$, the bi-PIKAS for $(\lambda+l,\nu+l^{\vee})\in\operatorname{SC}(S)\times\operatorname{SC}(T)$ have the same underlying Kähler affine structure.

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  The bi-PIKAS for $(\epsilon^{-1}\lambda,\epsilon\nu)$ differs from the bi-PIKAS for $(\lambda,\nu)$ by the $\epsilon$-rescaling

  $$(y^{\prime}_{\alpha},\hat{y}^{\prime}_{\alpha},K^{\prime}_{\alpha}(y^{\prime}_{\alpha}),\hat{K}^{\prime}_{\alpha}(\hat{y}^{\prime}_{\alpha}),g_{ij})=(\epsilon^{-1}y_{\alpha},\epsilon\hat{y}_{\alpha},K_{\alpha}(\epsilon y^{\prime}_{\alpha}),\hat{K}_{\alpha}(\epsilon^{-1}\hat{y}^{\prime}_{\alpha}),\epsilon^{2}g_{ij})$$