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

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

Kragujevac Journal of Mathematics, Tome 35 (2011) no. 2, p. 249

Résumé

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.

Détail

Classification : 15A75 51N15 65D18 68U05 68U10
Keywords: Projective spherical space, Central projection in higher dimensions, Visibility algorithm, Non-Euclidean geometries by projective metrics