computational geometry

This work addresses the challenge of verifying spatial properties of polyhedral models (e.g., triangular/tetrahedral meshes). We propose an enhanced model-checking approach based on the spatial logic SLCS. Our method treats model-checking outcomes as atomic propositions fed back into the original model, enabling semantic enrichment and structural minimization. This feedback loop significantly reduces the length of subsequent spatial property specification formulas and uncovers implicit spatial structural features. Grounded in closure-space semantics, the approach is implemented within the PolyLogicA model checker and integrated with PolyVisualizer for intuitive, semantically faithful visualization. Experimental evaluation demonstrates that our technique preserves formal rigor while improving both the efficiency and interpretability of spatial analysis. It establishes a novel paradigm for automated verification and visualization in computer graphics and geometric modeling.

Existing Euclidean geometry instruction suffers from fragmented exercise organization and heavy reliance on manual feedback provision. Method: We propose an intelligent pedagogical framework integrating a revitalized classical geometry problem ontology with modern large language models (LLMs). Specifically, we (1) reconstruct and extend a three-decade-old geometry problem ontology to construct a structured solution graph enabling semantic exercise organization; and (2) design an ontology-constrained LLM reasoning and verification mechanism supporting automated solution-path generation, stepwise correctness assessment, and personalized feedback synthesis. Results: Empirical evaluation demonstrates high accuracy in solution verification and strong interpretability of generated feedback. Contribution: This work is the first to systematically embed a historical geometry ontology into LLM-based educational applications, establishing a principled integration pathway between legacy educational resources and AI-powered teaching tools—thereby enabling scalable, verifiable, and pedagogically grounded support for both teacher-informed instruction and learner-centered practice.

This paper addresses the CG:SHOP 2025 challenge: computing constrained non-obtuse triangulations (all interior angles ≤ 90°) of planar domains, subject to mandatory inclusion of specified vertices and edges, while minimizing the number of Steiner points. We propose a local search framework built upon dynamically constrained Delaunay triangulation, integrating geometric conflict detection, adaptive Steiner point relocation, and a synergistic optimization strategy—simultaneously considering removal, relocation, and insertion of Steiner points. To our knowledge, this is the first approach achieving a Pareto balance between solution size and quality for this problem. Our method guarantees 100% compliance with the non-obtuse angle constraint. On diverse, complex benchmark instances, it reduces the average number of Steiner points by 37% compared to prior methods—marking a substantial improvement. The approach secured first place in the CG:SHOP 2025 competition.

This paper resolves the long-standing “minimum star-shaped partition of a simple polygon” problem—open since 1981—by covering a given simple polygon with the fewest non-overlapping star-shaped subpolygons, allowing Steiner points. The proposed method integrates geometric decomposition, visibility graph optimization, dynamic programming, and structural analysis of star kernels, constructing the DP state space over triangulations. It yields the first exact polynomial-time algorithm applicable to arbitrary simple polygons, overcoming prior restrictions to monotone or orthogonal polygons and eliminating the requirement to forbid Steiner points. The algorithm runs in O(n⁹) time, a substantial improvement over exponential brute-force approaches. This theoretical breakthrough enables direct applications in CNC pocket milling, motion planning, and shape parameterization, where minimal star-shaped decompositions are essential for efficient toolpath generation, collision-free navigation, and domain mapping.

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.

Traditional program semantic graphs struggle to uniformly represent the multi-way colocation, routing constraints, and higher-order product structures of geometric algebra inherent in spatial computing, thereby hindering the co-optimization of compiler transformations and physical correctness. This work proposes the Program Hypergraph (PHG), which generalizes binary edges to arbitrary-arity hyperedges and formally introduces hypergraphs into compiler intermediate representations for the first time. PHG naturally encodes mesh topology and Clifford algebra gradings via k-simplices. Integrated with an Abelian group–based dimensional type system (DTS), PHG enables unified reasoning—within a single structure—over geometric correctness, memory layout, precision selection, and hardware partitioning. This framework achieves end-to-end physics-aware compilation with automatic sparsity inference, eliminating the need for manual specialization and significantly enhancing both performance and portability of geometric algebra neural networks.

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.

This work proposes an end-to-end approach based on Satisfiability Modulo Theories (SMT) for automatically generating maze paths from arbitrary input patterns—such as text or shapes—that satisfy prescribed topological constraints, with support for both two-dimensional and three-dimensional physical realizations. The method formalizes path synthesis as a global constraint satisfaction problem encompassing adjacency, continuity, and pattern coverage, enabling the generation of self-avoiding planar paths or layered paths with specified over-and-under crossings through a single SMT solving step. To the best of our knowledge, this is the first application of SMT-based formal methods to pattern-driven maze generation, unifying path synthesis and 2D/3D geometric construction. The framework successfully translates textual or graphical inputs into fabricable maze structures, validated through a complete pipeline and demonstrated via representative examples.

This study addresses the problem of partitioning a polygon into the minimum number of strips of width at most 1, aligned with a given orthogonal direction, and producing a compact representation of the optimal partition. It introduces, for the first time in this domain, the Clarke–Cormack–Burkowski lattice-theoretic framework, modeling the problem via interval antichains and combining meet/join operations with dynamic programming to devise an input-sensitive optimal algorithm. For convex polygons, the approach achieves an O(log n)-time decision version and an O(h log(1 + n/h))-time reporting version, where h is the number of strips in the optimal solution. For both simple and self-overlapping polygons, it attains O(n log n) time complexity, while establishing matching lower bounds of Ω(n) and Ω(n log n), respectively, thereby yielding tight complexity characterizations for all three polygon classes.

This work addresses the inefficiency of spatial model checking on large-scale images, particularly the high computational cost associated with quasi-discrete closure models. To overcome this challenge, the authors propose an efficient approach that encodes spatial closure models as labeled transition systems and introduces, for the first time, a branching bisimulation minimization algorithm to compute CoPa equivalence classes. This technique preserves the correspondence between pixels and their equivalence classes while substantially accelerating detection. The resulting method constitutes a sound and efficient CoPa bisimulation minimization framework, validated through the VoxMinX prototype toolchain on real-world large-scale images, demonstrating significant speedup in spatial property model checking.