Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2603.16488

A Complete Graphic Statics for Rigid-Jointed 3D Frames. Part 3: Loops for Kinematics

Allan McRobie
View original source ↗Read source document ↗Export metadata ↓

Abstract

In Part 3 of this sequence of papers, the kinematic behaviour of 3D frame structures is described using the loop formalism that was developed in Part 2 to describe equilibrium. There, the notions of polygons, polyhedra and polytopes that form the geometric toolbox underlying graphic statics were replaced by the more general concept of CW-complexes from algebraic homology. The six components of the stress resultant acting on any cut face of a bar in a rigidly-jointed framework were represented by the oriented bivector areas of the six projections of a loop in a 4D-space, with three components representing the force and three components representing the moment. In this paper, projected areas of loops in 4D will represent kinematic variables, with three projected areas representing the displacement of a point on the frame, and three other projected areas representing the rotation of the structure at that point. The 4D setting for the theory consists of the usual three dimensions of physical space together with a fourth dimension for the stress function. Virtual Work then manifests as a top form (an oriented 4-volume) in this 4D setting, being the integral over the structure of the wedge product of bivectors representing the local equilibrium and kinematic variables.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Allan McRobie. 2026-03-17. A Complete Graphic Statics for Rigid-Jointed 3D Frames. Part 3: Loops for Kinematics. https://arxiv.org/abs/2603.16488

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A positive solution to the $L^p$ projection centroid conjecture

In a classical paper [21] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [21], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

math.MG↗
arXiv2026

Finite and Dynamic Stability Horizons for Nearest-Neighbor Future Structures

Nearest-neighbor graphs are discrete objects whose membership may change under small perturbations of the underlying coordinates. We establish an explicit stability guarantee for finite labeled configurations in an arbitrary metric space. If the gap between the kth and (k+1)st distances is positive, simultaneous per-label perturbations smaller than one quarter of that gap preserve the directed k-nearest-neighbor membership. The factor four is sharp under the stated uniform displacement assumptions, and every fixed deterministic construction based solely on the labeled neighbor family is consequently invariant. Under an interval-valid Lipschitz bound for labeled information-space trajectories, the same result yields a certified lower bound on the first possible rewiring time, with a refinement for label-specific motion bounds. We apply the finite theorem to frozen standardized Taylor-Green future-information coordinates [d_B, log A_B] for 585 particles at three observed time strata. Outward-rounded interval arithmetic certified a sufficient perturbation radius of approximately 1.023 x 10^-6, and a separately implemented checker within the same research workflow verified the rank and distance-margin calculations. An outcome-blind audit then examined whether the construction supported a numerical continuous-time horizon. Because the complete information map involved discrete clustering and boundary reconstruction and lacked an analytic derivative bound, validated dense-time bound, or certified modulus of continuity, the application was correctly classified as DISCRETE_ONLY with STOP_NO_INTERVAL_VALID_BOUND. Thus finite local structural invariance is proved and numerically certified for the frozen configuration, while temporal specialization and predictive generalization remain separate questions requiring additional evidence.

math.MG↗
arXiv2026

Equivalence of Landscape and Erosion Distances for Persistence Diagrams

This paper establishes connections between three of the most prominent metrics used in the analysis of persistence diagrams in topological data analysis: the bottleneck distance, Patel's erosion distance, and Bubenik's landscape distance. Our main result shows that the erosion and landscape distances are equal, thereby bridging the former's natural category-theoretic interpretation with the latter's computationally convenient structure. The proof utilizes the category with a flow framework of de Silva et al., and leads to additional insights into the structure of persistence landscapes. Our equivalence result is applied to prove several results on the geometry of the erosion distance. We show that the erosion distance is not a length metric, and that its intrinsic metric is the bottleneck distance. We also show that the erosion distance does not coarsely embed into any Hilbert space, even when restricted to persistence diagrams arising from degree-0 persistent homology. Moreover, we show that erosion distance agrees with bottleneck distance on this subspace, so that our non-embeddability theorem generalizes several results in the recent literature.

math.MG↗