On Linear Spaces of Polyhedral Meshes

Area of Science:

• Computer Graphics
• Computational Geometry
• Geometric Modeling

Background:

• Polyhedral meshes (PM) are increasingly vital in architectural and industrial design.
• Generating and manipulating PMs is challenging with existing surface approximation methods.

Purpose of the Study:

• To develop a reverse approach for polyhedral mesh generation.
• To explore the space of possible planar meshes given a specific topology.
• To provide tools for global and local design of polyhedral meshes.

Main Methods:

• Characterizing maximal linear spaces of polyhedral meshes within a given topology.
• Describing these linear spaces as nullspaces of differential operators.
• Analyzing the operator to understand geometric possibilities and limitations.

Main Results:

• A complete characterization of linear spaces of polyhedral meshes.
• Demonstration that these spaces are nullspaces of differential operators.
• Development of design tools based on operator analysis.

Conclusions:

• The reverse approach offers a new perspective on polyhedral mesh generation.
• The differential operator framework provides insights into mesh design.
• This method enhances the understanding of geometric possibilities and constraints for given topologies.