A Unified Framework for N-Dimensional Visualization and Simulation: Implementation and Evaluation including 4D Boolean

TL;DR

This work tackles the lack of integrated, interactive environments for high-dimensional visualization by introducing a CPU-based framework that unifies Quickhull-based convex hulls, hyperplane slicing for visualization, and N-dimensional Boolean operations, complemented by an XPBD-based 4D physics demonstration. It emphasizes algorithmic transparency and separation of topology and geometry, implemented in Unity/C# with a modular, extensible architecture and the Plex data format for data exchange. Key contributions include the Direct Quickhull approach for hull construction, hierarchical cross-sectioning for visualization, a robust N-D Boolean pipeline, and a dual JSON/.plex ecosystem, all demonstrated through 4D experiments. This framework lowers entry barriers for high-dimensional research and has practical implications for education and entertainment, while outlining future work on higher dimensions and GPU-accelerated implementations.

Abstract

This study proposes a unified framework for simulation and visualization of intuitive exploration of phenomena in N-dimensional space. While specialized libraries offer powerful geometric algorithms, they typically lack integrated environments for interactive trial and error, creating a barrier for researchers. The contribution of this research is the integration of Quickhull-based mesh generation, visualization via hyperplane slicing, and computationally expensive Boolean operations into a single, extensible platform, while maintaining interactivity. To validate its effectiveness, this paper presents a 4-dimensional implementation and introduces a new interaction design, termed `High-Dimensional FPS,' to enable intuitive high-dimensional exploration. Furthermore, as a case study to demonstrate the framework's high extensibility, I also integrated a non-rigid body physics simulation based on Extended Position Based Dynamics (XPBD). Experimental results confirmed the effectiveness of the proposed method, achieving real-time rendering (80 fps) of complex 4D objects and completing Boolean operations within seconds in a standard PC environment. By providing an accessible and interactive platform, this work lowers the entry barrier for high-dimensional simulation research and enhances its potential for applications in education and entertainment.

Figures (9)

• Figure 1: An overview of the proposed framework. The left side shows the static shape definition workflow, and the right side shows the dynamic visualization pipeline.
• Figure 2: Stages of vertex management and convex hull construction in Direct Quickhull. Orange vertices are unprocessed, gray vertices are processed, gray facets are interior, and black facets form the outer hull. In the iterative step, the convex hull is expanded using a selected facet from the hull (red) and its farthest point (red).
• Figure 3: The logic for determining visible facets in Quickhull. This illustrates the method for identifying facets that are visible from the farthest point $\mathbf{P}_{\text{new}}$. The centroid of the facet being tested is denoted as $\mathbf{P}_{\text{c}}$, and its normal vector is $\mathbf{n}$. The vector pointing from $\mathbf{P}_{\text{new}}$ to $\mathbf{P}_{\text{c}}$ is defined as $\mathbf{v}$.
• Figure 4: The sequential tessellation process of a facet in a Boolean operation (analogized in 3D). After the input facet (yellow) is clipped and tessellated by a facet from the opposing object (blue, left), each resulting facet is further tessellated by a subsequent blue facet, as shown in the center. By repeating this process for all intersecting facets, a mesh (Output) that is accurately partitioned at the boundary with the opposing object is ultimately generated.
• Figure 5: Classification of general and degenerate intersection cases. This figure categorizes the general case, where a ray crosses a single facet, and representative degenerate cases that require precise intersection tests, where the ray passes exactly through a facet, edge, or vertex.