In this paper, we present a mathematical visualization paradigm for exploring curves embedded in 3D and surfaces in 4D mathematical world. The basic problem is that, 3D figures of 4D mathematical entities often twist, turn, and fold back on themselves, leaving important properties behind the surface sheets. We propose an interactive system to visualize the topological features of the original 4D surface by slicing its 3D figure into a series of feature diagram. A novel 4D visualization interface is designed to allow users to control 4D topological shapes via the collection of diagram handles using the established curve manipulation mechanism. Our system can support rich mathematical interaction of 4D mathematical objects which is very difficult with any existing approach. We further demonstrate the effectiveness of the proposed visualization tool using various experimental results and cases studies.

We present KnotPad, an interactive paper-like system for visualizing and exploring mathematical knots; we exploit topological drawing and math-aware deformation methods in particular to enable and enrich our interactions with knot diagrams. Whereas most previous efforts typically employ physically based modeling to simulate the 3D dynamics of knots and ropes, our tool offers a Reidemeister move based interactive environment that is much closer to the topological problems being solved in knot theory, yet without interfering with the traditional advantages of paper-based analysis and manipulation of knot diagrams. Drawing knot diagrams with many crossings and producing their equivalent is quite challenging and error-prone. KnotPad can restrict user manipulations to the three types of Reidemeister moves, resulting in a more fluid yet mathematically correct user experience with knots. For our principal test case of mathematical knots, KnotPad permits us to draw and edit their diagrams empowered by a family of interactive techniques. Furthermore, we exploit supplementary interface elements to enrich the user experiences. For example, KnotPad allows one to pull and drag on knot diagrams to produce mathematically valid moves. Navigation enhancements in KnotPad provide still further improvement: by remembering and displaying the sequence of valid moves applied during the entire interaction, KnotPad allows a much cleaner exploratory interface for the user to analyze and study knot equivalence. All these methods combine to reveal the complex spatial relationships of knot diagrams with a mathematically true and rich user experience.

An efficient approach to recover implicit sharp edges with the help of filling holes in meshes is presented in this paper. First, the edges near the hole are classified as smooth and sharp edges by the dihedral angel between their adjacent triangles. The hole-nodes are classified as smooth and sharp nodes according to the types of their adjacent edges. Second, different methods are adopted to shrinkage the holes for different types of hole-nodes. A triangulation method is used for smooth hole-nodes, while an extension method is used for sharp hole-nodes. Third, the filled patch is refined for a smooth and uniform mesh preserving sharp edges logically.