A Flip‐book of Knot Diagrams for Visualizing Surfaces in 4‐Space
A Flip‐book of Knot Diagrams for Visualizing Surfaces in 4‐Space
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DOI：
10.1111/cgf.14545
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发表时间：
2022-06
影响因子：
2.5
通讯作者：
Huan Liu;Hui Zhang
中科院分区：
文献类型：
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作者：
Huan Liu;Hui Zhang
正如3D曲线的2D阴影在线交叉处失去结构一样，平滑4D拓扑表面的3D图形投影在一个表面与其自身相交处中断。它们扭曲、翻转、折叠，在表面层后面留下重要但隐藏的特征。在本文中，我们提出了一种智能切片工具，它可以在熵图中读取4D表面，并提出生成表面横截面图像（或“切片”）的最佳方法，以可视化其底层4D结构。我们的可视化将4D嵌入曲面视为按时间堆叠的3D曲线的集合，非常像翻页动画，其中序列中的连续项最多相差一个关键变化。这种新颖的方法可以生成拓扑意义的可视化来描绘复杂和不熟悉的4D表面，并具有最少数量的横截面图。我们的方法已成功用于创建图表翻页簿，以可视化一系列已知的4D表面。在这项初步研究中，我们的结果表明，新的可视化和切片工具可以帮助观众理解和描述复杂的空间关系和整体结构的4D表面。
Just as 2D shadows of 3D curves lose structure where lines cross, 3D graphics projections of smooth 4D topological surfaces are interrupted where one surface intersects itself. They twist, turn, and fold back on themselves, leaving important but hidden features behind the surface sheets. In this paper, we propose a smart slicing tool that can read the 4D surface in its entropy map and suggest the optimal way to generate cross‐sectional images — or “slices” — of the surface to visualize its underlying 4D structure. Our visualization thinks of a 4D‐embedded surface as a collection of 3D curves stacked in time, very much like a flip‐book animation, where successive terms in the sequence differ at most by a critical change. This novel method can generate topologically meaningful visualization to depict complex and unfamiliar 4D surfaces, with the minimum number of cross‐sectional diagrams. Our approach has been successfully used to create flip‐books of diagrams to visualize a range of known 4D surfaces. In this preliminary study, our results show that the new visualization and slicing tool can help the viewers to understand and describe the complex spatial relationships and overall structures of 4D surfaces.