论文标题
Apollonius表面，四面体，Menelaus'和Ceva定理的限制面，$ \ sxr $和$ \ hxr $几何形状
Apollonius surfaces, circumscribed spheres of tetrahedra, Menelaus' and Ceva's theorems in $\SXR$ and $\HXR$ geometries
论文作者
论文摘要
在本文中，我们研究了$ \ sxr $和$ \ hxr $几何形状，它们是均匀的瑟斯顿3晶格。我们定义并确定广义的Apollonius表面，并通过它们定义“大地三角形的表面”。使用上述Apollonius表面，我们开发了一个过程，以确定任意$ \ sxr $和$ \ hxr $ tetrahedron的限制的中心和半径。此外，我们概括了著名的Menelaus和Ceva的定理，用于两个空间中的大地三角形。在我们的工作中，我们将使用$ \ sxr $的投影模型和$ \ hxr $的几何形状，由E.Molnár在\ cite {m97}中描述。
In the present paper we study $\SXR$ and $\HXR$ 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 $\SXR$ and $\HXR$ tetrahedron. Moreover, we generalize the famous Menelaus' and Ceva's theorems for geodesic triangles in both spaces. In our work we will use the projective model of $\SXR$ and $\HXR$ geometries described by E. Molnár in \cite{M97}.
