Visualizing 2-dimensional Manifolds with Curve Handles in 4D

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.