关于欧氏空间中的循环四元数和hyperbolic geometries
On cyclic quadrilaterals in euclidean and hyperbolic geometries
摘要：
在单位圆上按正序排列的四个点确定四边形的顶点，四边形被认为是欧几里得四边形或双曲四边形，这取决于连接顶点的线是欧几里得线还是双曲线。在双曲线的情况下，这种类型的四边形称为理想四边形。我们的主要结果给出了一个欧几里德对应的早期结果的双曲距离之间的对边的理想四边形。证明是基于计算涉及双曲几何。我们还得到了单位圆盘内双曲测地线线段的双曲中点的一个新公式。作为某些几何性质的应用，我们给出了单位圆上任意四点关于一个直径对称化的欧氏构造，它保持了四元组的绝对交比。
英文摘要：
Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are euclidean or hyperbolic lines. In the case of hyperbolic lines, this type of quadrilaterals are called ideal quadrilaterals. Our main result gives a euclidean counterpart of an earlier result on the hyperbolic distances between the opposite sides of ideal quadrilaterals. The proof is based on computations involving hyperbolic geometry. We also found a new formula for the hyperbolic midpoint of a hyperbolic geodesic segment in the unit disk. As an application of some geometric properties, we provided a euclidean construction of the symmetrization of random four points on the unit circle with respect to a diameter which preserves the absolute cross ratio of quadruples.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
