Title: Stereographic Projections on Some Quadric Surfaces Show Chinese title

Title: 一些二次曲面的 stereographic 投影

Abstract: In this work, we present an adaptation of the classical stereographic projection, originally formulated for the sphere, now considering the context of the ellipsoid and the elliptic paraboloid. We begin by constructing the stereographic projections for both quadric surfaces separately, analyzing the geometric particularities of each surface and the challenges arising from their variable curvatures and, in the case of the paraboloid, its non-compactness. In the final part of the work, we establish results concerning the eccentricities, curvatures, arc length, and areas of the ellipses formed by the intersection of the quadrics with horizontal sections and their corresponding projections onto the plane-xy. Show Chinese abstract

Abstract: 在这项工作中，我们提出了经典 stereographic 投影的一个适应版本，该投影最初是为球面设计的，现在考虑椭球面和椭圆抛物面的背景。 我们首先分别构造这两个二次曲面的 stereographic 投影，分析每个曲面的几何特性及其由于可变曲率（对于抛物面还包括非紧性）带来的挑战。 在工作的最后部分，我们研究了由二次曲面与水平截面相交形成的椭圆的偏心率、曲率、弧长以及平面-xy 上相应投影的面积等相关结果。