arXiv subjects

Cameron Millar

Publications and source records attributed to Cameron Millar.

5 recordsLinked to original sources

arXiv2026

On weavings, grillages, tensegrities, and frameworks

We investigate the stability of discrete structures comprising of woven elastic beams in a plane, exposing a connection between admissible over / under patterns and the first-order and static rigidity of an associated tensegrity that is related through a natural polarity transformation. The relationship between the Airy and Whiteley stress functions for the tensegrity and weaving structures is explored. Our results lead to an efficient method for finding such an over / under pattern. The method is illustrated through a worked example modelled on the complete bipartite graph $K_{4,4}$.

math.MG↗

arXiv2025

Equilibrium stresses in frameworks via symmetric averaging

For a bar-joint framework $(G,p)$, a subgroup $\Gamma$ of the automorphism group of $G$, and a subgroup of the orthogonal group isomorphic to $\Gamma$, we introduce a symmetric averaging map which produces a bar-joint framework on $G$ with that symmetry. If the original configuration is ``almost symmetric", then the averaged one will be near the original configuration. With a view on structural engineering applications, we then introduce a hierarchy of definitions of ``localised" and ``non-localised" or ``extensive" self-stresses of frameworks and investigate their behaviour under the symmetric averaging procedure. Finally, we present algorithms for finding non-degenerate symmetric frameworks with many states of self-stress, as well as non-symmetric and symmetric frameworks with extensive self-stresses. The latter uses the symmetric averaging map in combination with symmetric Maxwell-type character counts and a procedure based on the pure condition from algebraic geometry. These algorithms provide new theoretical and computational tools for the design of engineering structures such as gridshell roofs.

math.MG↗

arXiv2024

ARSecure: A Novel End-to-End Encryption Messaging System Using Augmented Reality

End-to-End Encryption (E2EE) ensures that only the intended recipient(s) can read messages. Popular instant messaging (IM) applications such as Signal, WhatsApp, Apple's iMessage, and Telegram claim to offer E2EE. However, client-side scanning (CSS) undermines these claims by scanning all messages, including text, images, audio, and video files, on both sending and receiving ends. Industry and government parties support CSS to combat harmful content such as child pornography, terrorism, and other illegal activities. In this paper, we introduce ARSecure, a novel end-to-end encryption messaging solution utilizing augmented reality glasses. ARSecure allows users to encrypt and decrypt their messages before they reach their phone devices, effectively countering the CSS technology in E2EE systems.

cs.CR↗

arXiv2024

Homology of Moment Frames

Using homological techniques we show that a pin-anchored frame that involves only moments and shears provides a conceptual bridge between the statics of moment frames and the kinematics of pin-jointed trusses. One immediate result is a long exact sequence whose alternating sum of dimensions gives a novel counting rule for self-stresses and mechanisms. This combines the Maxwell-Calladine count for pin-jointed trusses with the circuit rank (first Betti number) associated with self-stresses in moment frames. These relations apply to frames in 2, 3 or any dimensions. This work heralds a shift towards a deeper study of the relationships and dualities that exist between structural equilibria and kinematics.

math.AT↗

arXiv2021

States of self-stress in symmetric frameworks and applications

We use the symmetry-extended Maxwell rule established by Fowler and Guest to detect states of self-stress in symmetric planar frameworks. The dimension of the space of self-stresses that are detectable in this way may be expressed in terms of the number of joints and bars that are unshifted by various symmetry operations of the framework. Therefore, this method provides an efficient tool to construct symmetric frameworks with many `fully-symmetric' states of self-stress, or with `anti-symmetric' states of self-stress. Maximizing the number of independent self-stresses of a planar framework, as well as understanding their symmetry properties, has important practical applications, for example in the design and construction of gridshells. We show the usefulness of our method by applying it to some practical examples.

math.MG↗