Second-order prestress stability and third-order rigidity of polyhedral surfaces

TL;DR

This paper introduces new criteria for second-order prestress stability and discusses third-order rigidity limitations in polyhedral surfaces, with broader implications for geometric constraint systems.

Contribution

It develops an energy-based approach to analyze second-order prestress stability and clarifies the conditions under which third-order rigidity can be tested.

Findings

New criterion for second-order prestress stability

Limitations identified for testing third-order rigidity

Applicability extends to general geometric constraint systems

Abstract

The energy-based definition provides a viable resolution to the longstanding confusion on the proper definition of $n$-th order rigidity and flexibility in geometric constraint systems. Applying an energy-based local rigidity analysis to polyhedral surfaces, we derive new criterion for testing second-order prestress stability. Additionally, we discuss the limitations of testing third-order rigidity, except in the special case where the null space of the rigidity matrix is one-dimensional. Although our primary focus is on polyhedral surfaces, the theory and methods developed in this article apply to general geometric constraint systems under certain conditions.