4D Data Visualization Using Iso-surfaces and a Control Plane Lutz Kettner * Jarek Rossignac † Jack Snoeyink * Introduction Many areas of science and engineering rely on our ability to efficiently analyze large data sets from a four-dimensional (4d) space-time domain. These data sets represent the evolution through time of quantities such as temperature, pressure, and flow velocity that are measured or computed on a dense set of sample points in three-dimensional (3d) space. At a high-performance computing center (HPCC), full-scale simulations of atmospheric turbulent flow or a combustion in an engine produce hundreds of gigabytes of data. Often, preselected 2d slices or 2d iso-surfaces in 3d are downloaded from the HPCC to visualize such data sets; if a researcher suspects that a different slice will show a phenomenon of interest, he or she must download and manipulate data locally, or rerun the entire simulation. Researchers would bene- fit from an efficient interactive remote access to sim- ulation data, but limited bandwidth is a bottleneck. Increases in bandwidth, processing, and storage are not likely to eliminate this problem because they also increase the size of data sets produced. We are investigating a client-server architecture for interactive remote analysis based on partitioning the data dimensions. A visualization server stores data on an adaptively simplified 4d mesh of the domain, and a client views an iso-surface in a 3d volume by selecting a particular time and iso-value. Time and iso-value form a 2d-plane, or control plane, which represents the space of all iso-surfaces. The system can color points of the control plane with informa- tion computed for corresponding iso-surfaces. As the user navigates the data by small changes in time or iso-value, the system can exploit coherence to update the iso-surface at the client. We have begun to explore several issues and trade- offs for such a system: from data structures and algo- rithms to modes of user interaction. In this abstract, we report on initial feasibility studies for displaying topological properties of iso-surfaces at corresponding points of the control plane. We begin by describing the splitting of dimensions between iso-surface and * University of North Carolina at Chapel Hill † Georgia Institute of Technology control plane when the server’s data structure is a mesh of 4-simplices or pentatopes. Iso-Surfaces from a Pentatope Mesh Suppose that pressure is the variable defining the iso- surface. The server keeps a mesh of pentatopes over the 4d domain, with some general position assump- tions. The pressure p is given for each vertex and is linearly interpolated in the interior of each pentatope. Thus, we have a function p = f (x,y,z,t) defined over the domain. The function f defines a terrain in di- mension five: {(x, y, z, t, p) ∈ R 5 | p = f (x,y,z,t)}. We obtain an iso-surface by restriction along two di- mensions, pressure p and time t: iso (p, t) := {(x, y, z) ∈ R 3 | f (x,y,z,t)= p} The Marching Cubes algorithm [3] has become a stan- dard tool for iso-surface extraction in 3d. A simple variant, often called Marching Tetrahedra, applies to pentatope meshes and is free of topological ambigui- ties. (See, e.g., [2].) With this notation, the control plane is simply the pt-plane. Iso-surface iso (p, t) intersects a pentatope, π, if and only if the point (p, t) is inside the projec- tion of π on the control plane. Moreover, we may determine the surface portion s = π ∩ iso (p, t), which is the intersection in xyz-space between pentatope π and iso-surface iso (p, t), from this projection. 3 4 5 3 4 5 3 4 5 Figure 1: Three projections of pentatopes into the pt-plane. Numbers indicate covering triangles. We have the following correspondences between the pentatope π and surface s, because we slice away the two dimensions p and t to form s. Since π is a convex 4d set in 5d, s is a convex 2d set in 3d. Boundary tetrahedra (3d) and triangles (2d) of π contribute boundary edges (1d) and vertices (0d) of s. As in 1