Visualizing and Slicing Topological Surfaces in Four Dimensions

Abstract

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.