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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. 

Extracting good views from a large sequence of visual frames is quite difficult but a very important task across many fields. Fully automatic view selection suffers from high data redundancy and heavy computational cost, thus fails to provide a fast and intuitive visualization. In this paper we address the automatic viewpoint selection problem in the context of 3D knot deformation. After describing viewpoint selection criteria, we detail a brute-force algorithm with a minimal distance alignment method in a way to not only ensure the global best viewpoint but also present a sequence of visually continuous frames. Due to the intensive computation, we implement an efficient extraction method through parallelization. Moreover, we propose a fast and adaptive method to retrieve best viewpoints in real-time. Despite its local searching nature, it is able to generate a set of visually continuous key frames with an interactive rate. All these combine provide insights into 3D knot deformation where the critical changes of the deformation are fully represented. 

Geometric problems of interest to mathematical visualization applications involve changing structures, such as the moves that transform one knot into an equivalent knot. In this paper, we describe mathematical entities (curves and surfaces) as link-node graphs, and make use of energy-driven relaxation algorithms to optimize their geometric shapes by moving knots and surfaces to their simplified equivalence. Furthermore, we design and conFigure parallel functional units in the relaxation algorithms to accelerate the computation these mathematical deformations require. Results show that we can achieve significant performance optimization via the proposed threading model and level of parallelization. 

We present a computer interface to visualize and interact with mathematical knots, i.e., the embeddings of closed circles in 3-dimensional Euclidean space. Mathematical knots are slightly different than everyday knots in that they are infinitely stretchy and flexible when being deformed into their topological equivalence. In this work, we design a visualization interface to depict mathematical knots as closed node-link diagrams with energies charged at each node, so that highly-tangled knots can evolve by themselves from high-energy states to minimal (or lower) energy states. With a family of interactive methods and supplementary user interface elements, out tool allows one to sketch, edit, and experiment with mathematical knots, and observe their topological evolution towards optimal embeddings. In addition, out interface can extract from the entire knot evolution those key moments where successive terms in the sequence differ by critical change; this provides a clear and intuitive way to understand and trace mathematical evolution with a minimal number of visual frames. Finally out interface is adapted and extended to support the depiction of mathematical links and braids, whose mathematical concepts and interactions are just similar to our intuition about knots. All these combine to show a mathematically rich interface to help us explore and understand a family of fundamental geometric and topological problems. 

Mathematical knots are different from everyday ropes in that they are infinitely stretchy and flexible when being deformed into their ambient isotopic. For this reason, a number of challenges arise when visualizing mathematical knot's static and changing structures during topological deformation. In this paper we focus on computational methods to visually communicate the mathematical knot's dynamics by computationally simulating the topological deformation and capturing the critical changes during the entire simulation. To further improve our visual experience, we propose a fast and adaptive method to extract key moments where only critical changes occur to represent and summarize the long deformation sequence. We conduct evaluation study to showcase the efficacy and efficiency of our methods.