论文标题
HOPF振动的合成构造在4空间的双正交投影中
Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space
论文作者
论文摘要
拓扑师众所周知，在3杆上到2个球上的点上的HOPF纤维映射圆圈是2个球面上的点。虽然将2个球员嵌入3个空间中，但需要四维欧几里得空间才能可视化3个球体。使用计算机图形学基于它们的分析表示，在4空间中可视化对象已在近几十年来流行。对于纯合成构建体，我们将最近引入的4空间可视化方法通过其双向异构投影对两个相互垂直的3个空间进行了研究，以研究HOPF纤维作为四维关系，而在较低的尺寸中没有类比。在本文中，双向共同投影的方法用于从2个球体上的相应点的3杆纤维的直接合成结构。 2个球面上的大圆圈的纤维创建了在造型3空间上以立体投影可视化的嵌套托里。逐步结构是由动态的三维模型补充的，同时显示了纤维和相互关系的三个球，2杆和刻度图。合成结构的每个步骤都由其分析表示支持，以突出两种解释之间的联系。
The Hopf fibration mapping circles on a 3-sphere to points on a 2-sphere is well known to topologists. While the 2-sphere is embedded in 3-space, four-dimensional Euclidean space is needed to visualize the 3-sphere. Visualizing objects in 4-space using computer graphics based on their analytical representations has become popular in recent decades. For purely synthetic constructions, we apply the recently introduced method of visualization of 4-space by its double orthogonal projection onto two mutually perpendicular 3-spaces to investigate the Hopf fibration as a four-dimensional relation without analogy in lower dimensions. In this paper, the method of double orthogonal projection is used for a direct synthetic construction of the fibers of a 3-sphere from the corresponding points on a 2-sphere. The fibers of great circles on the 2-sphere create nested tori visualized in a stereographic projection onto the modeling 3-space. The step-by-step construction is supplemented by dynamic three-dimensional models showing simultaneously the 3-sphere, 2-sphere, and stereographic images of the fibers and mutual interrelations. Each step of the synthetic construction is supported by its analytical representation to highlight connections between the two interpretations.
