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Smooth topological surfaces embedded in 4D create complex internal structures in their projected 3D figures. Often these 3D figures twist, turn, and fold back on themselves, leaving important properties behind the surface sheets. Triangle meshes are not well suited for illustrating such internal structures and their topological features. In this paper, we propose a new approach to visualize these internal structures by slicing the 4D surfaces in our dimensions and revealing the underlying 4D structures using their cross-sectional diagrams. We think of a 4D-embedded surface as a collection of 3D curves stacked and evolved in time, very much like a 3D movie in a time-elapse form; and our new approach is to translate a surface in 4-space into such a movie — a sequence of time-lapse frames where successive terms in the sequence differ at most by a critical change. The visualization interface presented in this paper allows us to interactively define the longitudinal axis, and the automatic algorithms can partition the 4D surface into parallel slices and expose its internal structure by generating a time-lapse movie consisting of topologically meaningful cross-sectional diagrams from the representative slices. We have extracted movies from a range of known 4D mathematical surfaces with our approach. The results of the usability study show that the proposed slicing interface allows a mathematically true user experience with surfaces in four dimensions.
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A Flip‐book of Knot Diagrams for Visualizing Surfaces in 4‐Space
Abstract 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.
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- Award ID(s):
- 1651581
- PAR ID:
- 10406067
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Computer Graphics Forum
- Volume:
- 41
- Issue:
- 3
- ISSN:
- 0167-7055
- Format(s):
- Medium: X Size: p. 345-354
- Size(s):
- p. 345-354
- Sponsoring Org:
- National Science Foundation
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