In this paper, we show the use of visualization and topological relaxation methods to analyze and understand the underlying structure of mathematical surfaces embedded in 4D. When projected from 4D to 3D space, mathematical surfaces often twist, turn, and fold back on themselves, leaving their underlying structures behind their 3D figures. Our approach combines computer graphics, relaxation algorithm, and simulation to facilitate the modeling and depiction of 4D surfaces, and their deformation toward the simplified representations. For our principal test case of surfaces in 4D, this for the first time permits us to visualize a set of well-known topological phenomena beyond 3D that otherwise could only exist in the mathematician’s mind. Understanding a fairly long mathematical deformation sequence can be aided by visual analysis and comparison over the identified “key moments” where only critical changes occur in the sequence. Our interface is designed to summarize the deformation sequence with a significantly reduced number of visual frames. All these combine to allow a much cleaner exploratory interface for us to analyze and study mathematical surfaces and their deformation in topological space.

中文翻译：

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四维松弛拓扑表面

在本文中，我们展示了使用可视化和拓扑松弛方法来分析和理解嵌入在 4D 中的数学曲面的基础结构。当从 4D 空间投影到 3D 空间时，数学表面通常会扭曲、转动和向后折叠，将其底层结构留在 3D 图形后面。我们的方法结合了计算机图形学、松弛算法和模拟，以促进 4D 表面的建模和描绘，以及它们向简化表示的变形。对于我们 4D 表面的主要测试案例，这首次使我们能够将一组众所周知的 3D 拓扑现象可视化，否则这些现象只能存在于数学家的脑海中。通过对识别出的“关键时刻”进行视觉分析和比较，可以帮助理解相当长的数学变形序列，其中仅在序列中发生关键变化。我们的界面旨在用显着减少的视觉帧数来总结变形序列。所有这些结合在一起，为我们提供了一个更清晰的探索界面，用于分析和研究数学曲面及其在拓扑空间中的变形。