The present article provides a new characterization of the geometry on the points and hyperbolic lines of a non-degenerate symplectic polar space. This characterization is accomplished by studying the family of subspaces obtained when considering the polars of all hyperbolic lines. Let \(\mathbb W_{2n-1}(\mathbb F)\) denote the polar space with respect to a non-degenerate symplectic polarity of \(\mathbb P_{2n-1}(\mathbb F)\), for \(n\geqslant1\) and \(\mathbb F\) a field. The hyperbolic line graph \(\mathbf S(\mathbb W_{2n-1}(\mathbb F)) =\mathbf S_{2n-1}(\mathbb F)\) then is the graph on the hyperbolic lines of \(\mathbb W_{2n-1}(\mathbb F)\) where hyperbolic lines \(l\) and \(m\) are adjacent (in symbols \(l\bot m\)) iff all points of \(l\) are collinear (in \(\mathbb W_{2n-1}(\mathbb F)\)) to all points of \(m\). Let \(n\geqslant4\), and let \(\Gamma\) be a connected graph that is locally \(\mathbf S(\mathbb W_{2n-1}(\mathbb F))\). Then \(\Gamma\) is isomorphic to \(\mathbf S_{2n+1}(\mathbb F)\). A perp space is a partial linear space \((\mathcal P,\mathcal L)\) endowed with a symmetric relation \(\bot\subseteq\mathcal P\times\mathcal P\) such that for every point \(x\), whenever \(p\neq q\) are points on a line \(l\), the fact \(x\bot p\) and \(x\bot q\) implies \(x\bot y\) for all \(y\in l\). Let \(n\geqslant4\), and let \((\mathcal P,\mathcal L,\bot)\) be a perp space in which for any line \(k\in\mathcal L\) the space \(k^\bot\) is isomorphic to the hyperbolic symplectic geometry of \(\mathbb W_{2n-1}(\mathbb F)\) with \(l\bot m\) iff \(m\) is in the polar of \(l\) for (hyperbolic) lines \(l\), \(m\) inside \(k^\bot\). If the graph \((\mathcal L,\bot)\) is connected, then \((\mathcal P,\mathcal L,\bot)\) is isomorphic to the hyperbolic symplectic geometry \(\mathbb W_{2n+1}(\mathbb F)\).