Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S 2 × R and H 2 × R Geometries

Szirmai, Jenő (2022) Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S 2 × R and H 2 × R Geometries. QUARTERLY JOURNAL OF MATHEMATICS, 73 (2). pp. 477-494. ISSN 0033-5606

Preview

Text 2012.06155.pdf Download (685kB) | Preview

Abstract

In the present paper we study S(2)xR and H(2)xR geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the 'surface of a geodesic triangle'. Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesic sphere of an arbitrary S(2)xR and H(2)xR tetrahedron. Moreover, we generalize the famous Menelaus's and Ceva's theorems for geodesic triangles in both spaces. In our work we will use the projective model of S(2)xR and H(2)xR geometries described by E. Molnar in [6].

Actions (login required)

Edit Item