Higher-dimensional central projection into 2-plane with visibility and applications

J. Katona, E. Molnár, I. Prok, J. Szirmai

Applying $d$-dimensional projective spherical geometry $\mathcal{PS}^d(\mathbf{R},\mathbf{V}^{d+1},\boldsymbol{V}_{d+1})$, represented by the standard real $(d+1)$-vector space and its dual up to positive real factors as $\sim$ equivalence, the Grassmann algebra of $\mathbf{V}^{d+1}$ and of $\boldsymbol{V}_{d+1}$, respectively, represent the subspace structure of $\mathcal{PS}^d$ and of $\mathcal{P}^d$. Then the central projection from a $(d-3)$-centre to a $2$-screen can be discussed in a straightforward way, but interesting visibility problems occur, first in the case of $d = 4$ as a nice attractive application. So regular $4$-solids can be visualized in the Euclidean space $\mathbf{E}^4$ and non-Euclidean geometries, e.g. spherical $\mathbf{S}^4$ and hyperbolic $\mathbf{H}^4$ geometry. In a short report geodesics and geodesic spheres will also be illustrated in $\mathbf{H}^2\!\times\!\mathbf{R}$ and $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ spaces by projective metric geometry.