computational geometry implementation

The absence of formal verification methods for G-code linear motion in 3D printing hinders assurance of geometric-code consistency. Method: This paper proposes a dimension-elevated semantic representation framework that parses G-code into sets of axis-aligned bounding boxes and their approximated point clouds, integrating geometric modeling with program analysis to enable invariant checking and differential testing across the manufacturing pipeline. Contribution/Results: The framework supports, for the first time, quantitative cross-slicer comparison (Cura vs. PrusaSlicer), error localization, and root-cause analysis of defects introduced during mesh repair (e.g., in MeshLab or Meshmixer). Evaluated on 58 real-world models, it efficiently detects slicing anomalies induced by small geometric features, exposes behavioral discrepancies among mainstream slicers, and identifies new errors inadvertently introduced during repair—thereby significantly enhancing the verifiability and reliability of end-to-end additive manufacturing pipelines.

Recent progress in dynamic computational geometry has stagnated, primarily due to the lack of systematic frameworks for modeling both instantaneous and long-term geometric properties of moving point sets—contrasting sharply with well-established static paradigms. Method: This paper introduces a spatiotemporal trajectory-based algorithmic framework for dynamic geometry, integrating motion modeling, geometric evolution analysis, and complexity-aware optimization techniques. It systematically designs efficient algorithms for fundamental dynamic problems—including convex hulls, nearest neighbors, and Voronoi diagrams—under continuous motion. Contribution/Results: The work establishes the first comprehensive methodology for dynamic geometry, encompassing problem formalization, algorithm construction, and rigorous theoretical analysis. It significantly improves timeliness and scalability in maintaining geometric structures under dynamic updates. The framework provides a rigorous theoretical foundation and practical algorithmic tools for applications in autonomous driving, sensor networks, and spatiotemporal data analytics.

This paper addresses topological ambiguities and numerical robustness issues in Constructive Solid Geometry (CSG) Boolean operations and mesh repair—arising from non-manifold intersections, multi-operand expressions, and degenerate geometries (e.g., coplanar or collinear features). We present the first algorithm to construct an exact Weiler spatial decomposition model. Our method integrates exact geometric predicates (via multi-precision arithmetic), co-refinement, radial sorting, constrained Delaunay triangulation, and symbolic perturbation to achieve precise intersection localization, unambiguous face classification, and consistent regional subdivision. Key contributions include: (1) the first complete, exact implementation of the Weiler model; and (2) a unified geometric kernel architecture that systematically handles all degenerate cases, eliminating duplicate faces and topological inconsistencies. Evaluated on the Thingi10K and ThingiCSG benchmarks, our approach demonstrates significantly higher robustness than state-of-the-art methods.

Plane-based Geometric Algebra (PGA) suffers from low representational efficiency and limited expressiveness for modeling discrete geometric entities—specifically, k-simplices (e.g., vertices, edges, faces) and k-complexes (e.g., point clouds, line complexes, triangle meshes)—in computational geometry. Method: We propose a unified, compact PGA-based representation framework. Our approach introduces Euclidean and ideal norms to derive a dimension-agnostic, closed-form k-metric formula—unifying length, area, volume, and higher-dimensional measures—and enables coordinate-free computation of geometric quantities such as centroids and inertia tensors. By integrating the join operator, simplex decomposition, and linear combinations, the framework supports algebraic construction and manipulation of both k-simplices and k-complexes. Results: Experiments demonstrate significant efficiency and practicality in mesh processing tasks. The method establishes a scalable, coordinate-free geometric computing paradigm for high-dimensional discrete geometry modeling.

This work introduces, for the first time, a prediction-augmented algorithmic framework to computational geometry, specifically targeting the efficient computation of two-dimensional Delaunay triangulations (DT). Given a point set \( P \) and a predicted triangulation \( G \) that approximates the true DT, the paper proposes an adaptive correction algorithm based on an edge-difference metric \( D \), a violation measure \( d_{\text{vio}} \), and a randomized sampling probability \( \rho \). The main contributions include a deterministic algorithm running in \( O(n + D \log^3 n) \) time and an optimal randomized algorithm with expected time \( O(n + D \log n) \). Under a stochastic model combining edge inclusion and violation degrees, the approach further yields an almost-linear-time solution, which is also extended to related problems such as the Euclidean minimum spanning tree.

This study investigates which classical computational geometry problems can surpass the $O(n \log n)$ time lower bound when input points are pre-sorted along one- or two-dimensional coordinate axes. To this end, the paper introduces the “Presort Hierarchy” framework, offering the first systematic characterization of how presorted information influences the complexity of geometric problems. Leveraging randomized algorithms and complexity reductions, the authors establish that constructing quadtrees, Voronoi diagrams, Delaunay triangulations, and Euclidean minimum spanning trees belongs to the 2-Presortable class. They present expected-time algorithms with complexity $O(n\sqrt{\log n})$, substantially improving upon the traditional lower bound and resolving a long-standing open problem dating back to 1989.

This work formalizes and investigates the centrality problem in triangulations: given a set of input triangulations, it seeks a central triangulation that minimizes the sum of parallel flip distances to all inputs. As an NP-hard problem and the core challenge of the CG:SHOP 2026 algorithm competition, it is addressed through an integrated approach combining computational geometry, graph-theoretic modeling, and combinatorial optimization. Leveraging structural properties of parallel flips, the study designs efficient search and approximation algorithms. This paper presents the first systematic formulation of the problem, fostering interdisciplinary connections between discrete geometry and combinatorial optimization. By catalyzing diverse solution strategies through the competition, it establishes a practical benchmark for algorithmic performance in this emerging domain.

This work addresses the problem of parallel reconfiguration among multiple triangulations by selecting a central triangulation and computing short parallel flip sequences from each input triangulation to this center, with the objective of minimizing the total path length. To achieve this, the authors introduce a novel integration of SAT and MaxSAT solvers, proposing a specialized SAT encoding for bounded-length flip paths and a global optimization model for fixed path-length vectors. A greedy heuristic is further incorporated to accelerate the solution process. The method demonstrates outstanding performance on the CG:SHOP 2026 challenge benchmarks, securing first place by significantly improving both computational efficiency and solution quality.

This study addresses the high computational cost of verifying polytope vertex enumerations and the inefficiency of traditional formal methods. We propose a certificate-based formal verification framework that introduces an abstract simplicial complex generalizing normal fan triangulations as a novel completeness criterion. This approach reduces expensive numerical computations to membership tests and inexpensive combinatorial checks, substantially lowering verification complexity. The framework is implemented using the Rocq proof assistant with OCaml extraction. Experimental evaluations demonstrate certified speedups of 1.5× to 5× over lrslib across various polytopes, while rigorously establishing formal correctness guarantees.

Existing tools struggle to visualize higher-order Voronoi diagrams and Delaunay tessellations under polygonal metrics, particularly Hilbert geometry. This work proposes the first efficient, dynamically interactive visualization system that unifies the generation and display of arbitrary-order Voronoi diagrams, Delaunay tessellations, and their associated clustering, overlapping, and exterior structures under Hilbert, Funk, and Thompson polygonal metrics, leveraging computational geometry algorithms. The core contributions include an integrated framework for generating and interactively exploring higher-order Voronoi diagrams, the discovery that k-th order Voronoi cells need not be star-shaped, and the establishment of theoretical complexity bounds for the underlying algorithms.