Orthogonality relations for tetrahedra in elliptic and hyperbolic space. (English) Zbl 1302.51012

Summary: Two tetrahedra in Euclidean space are called orthologic if the four lines through the vertices of the first and perpendicular to the corresponding face planes of the second are concurrent. More generally, two tetrahedra are called skew-orthologic if these four lines lie in a regulus (they are skew generators of a hyperboloid). These relations are symmetric, that is, also the lines through vertices of the second and perpendicular to the corresponding face planes of the first tetrahedron are concurrent or lie in a regulus. Moreover, two tetrahedra are orthologic if and only if non-corresponding edges are orthogonal (they are “anti-orthogonal”). It is common to all results mentioned so far that they can be formulated exclusively in terms of incidence and orthogonality relations. This suggests to investigate them in non-Euclidean Cayley-Klein geometries where incidence is given by the underlying projective structure and orthogonality is replaced by polarity with respect to a quadric. We prove the symmetry of the defining condition of orthologic and skew-orthologic tetrahedra in these spaces. The theorems of orthogonal tetrahedra and orthogonal pairs of tetrahedra remain true and the notions of “anti-orthogonal” and “orthologic” still coincide in non-Euclidean geometry.


51M10 Hyperbolic and elliptic geometries (general) and generalizations
51N15 Projective analytic geometry
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