polygonal sdf computation

This work addresses the limitation of traditional signed distance functions (SDFs) in robot motion planning, where non-differentiability hinders gradient-based optimization. Focusing on convex polyhedra, the authors propose a novel closed-form differentiable SDF that replaces the non-smooth min-max operations in classical SDFs with Hölder minimum and maximum operators. This formulation preserves exact sign information while achieving global differentiability, without requiring iterative solvers and with native support for GPU parallelization. Experimental results demonstrate that the proposed method significantly outperforms existing approaches in computational efficiency and has been successfully deployed in real-time robotic arm control, confirming its practicality and effectiveness.

This work addresses the lack of intuitive pedagogical tools for understanding projective metrics—Thompson, Funk, anti-Funk, and Hilbert—in convex polygonal domains within computational geometry. We design and implement an interactive educational software system that, for the first time, uniformly supports dynamic ball construction, geodesic generation, and real-time visualization of all four metrics over arbitrary convex *n*-gons. The system integrates computational geometric algorithms, a parameterized path-traversal engine, and OpenGL/WebGL rendering. Its key innovation lies in transcending the Euclidean framework to enable exact computation and interactive exploration of metric balls under non-smooth boundary constraints. The software significantly enhances conceptual clarity in teaching and provides a reproducible, extensible experimental platform for advanced research, thereby filling a critical gap in both pedagogical resources and visualization tools for projective metric geometry.

This work addresses the challenge of real-time local planning in cluttered environments, where conventional optimization-based approaches suffer from high computational overhead due to frequent collision checks. The authors propose a geometrically exact polygonal signed distance function (PSDF) and construct a branch-free, weight-free tensorized geometry pipeline that enables efficient GPU batch computation and automatic differentiation. Integrated into a sequential quadratic programming framework, this pipeline yields PSDF-MPC, a real-time model predictive controller. Notably, the method achieves the first efficient GPU-parallel implementation of PSDF, decouples CPU and GPU workloads, and renders obstacle-avoidance constraint evaluation complexity independent of the number of obstacles. Experiments demonstrate that PSDF surpasses existing methods in both accuracy and efficiency for distance queries, while PSDF-MPC exhibits strong real-time performance and robust collision avoidance in both simulation and physical robot trials.

This work proposes the first differentiable geometry processing system that seamlessly integrates with modern machine learning frameworks, addressing the longstanding challenge of combining geometric algorithms—typically non-differentiable and reliant on complex control flow—with gradient-based optimization. By unifying the adjoint method with a scatter-gather mesh processing paradigm, the system enables efficient gradient computation for existing geometric algorithms without requiring algorithmic reimplementation. It supports state-of-the-art solvers such as local-global and ADMM schemes and provides native differentiability for classical operations including curvature flows and conformal parameterizations. Evaluated on multiple inverse geometry problems, the approach significantly reduces both memory consumption and computational overhead, outperforming general-purpose differentiable optimization tools in runtime efficiency while dramatically lowering implementation effort.

This work addresses the challenge of efficiently computing signed distances from arbitrary points to point clouds without explicit surface reconstruction or spatial discretization. It introduces a novel approach that leverages the torus as a local geometric primitive, exploiting its analytic distance function. A pretrained neural network predicts per-point curvature and offset parameters, enabling parallel, mesh-free construction of the signed distance field. The method uniquely unifies signed distance computation with classical reconstruction paradigms—such as winding number and Poisson surface reconstruction—and supports direct geometric operations including offsetting, Boolean combinations, and sphere tracing on point clouds generated from photogrammetry, meshes, 3D Gaussians, or neural implicit representations. This significantly enhances both the efficiency and practicality of downstream geometry processing tasks.

This work addresses the challenge of automatically formalizing and verifying proofs for International Mathematical Olympiad (IMO)-level Euclidean geometry problems. It introduces GeoFormalizer, a Mathlib-native agent framework that translates informal problem statements into a geometric intermediate representation (GeoIR) and then into Lean 4, iteratively refining the formalization through structural diagnosis and semantic evaluation. Complementing this, GeoProver generates lemmas via geometric proof planning and algebraizes subgoals, producing algebraic certificates using Singular or SymPy, all verified by the Lean kernel. The approach achieves the first large-scale, kernel-verified automated proofs and counterexample generation for IMO geometry, incorporating a counterexample-guided diagnostic mechanism that significantly enhances weak models’ formalization performance. On 43 historical IMO problems, it proves 29 directly, generates Lean-verified counterexamples for the remaining 14—later proving corrected versions—and establishes the largest automated, kernel-verified IMO geometry proof set to date, including 12 new proofs and formal refutations of two previously conjectured statements on Lean-IMO-Bench.

This work addresses the geometric inconsistency that arises when interpolating signed distance functions (SDFs) from discrete samples. To resolve this issue, the authors propose a hard geometric constraint framework grounded in the theoretical properties of SDFs, which for the first time rigorously guarantees that interpolated results are both compatible with the original data and correspond to realizable geometric surfaces. The method integrates an efficient greedy interpolation algorithm with GPU-accelerated preprocessing, enabling high-quality, geometrically consistent outputs across three representative tasks: global SDF refinement, mesh reconstruction, and pseudo-SDF correction. This approach overcomes a key limitation of existing techniques, which lack formal guarantees of geometric consistency.

本文介绍了一个开源Python包，用于解决几何函数理论中的极值问题，通过计算泰勒系数、Fekete-Szegő常数等，并提供不同级别的验证方法。

This study addresses the problem of finding the shortest path that sequentially visits a sequence of axis-aligned orthogonal polygons under the Manhattan metric. The work presents the first truly subquadratic-time algorithm for this problem in an orthogonal geometric setting, introducing near-linear (Õ(n)) and linear (O(n)) time solutions for orthogonally convex polygons and axis-aligned rectangles, respectively. Key technical contributions include the use of weighted Voronoi diagrams, rectangular decomposition, persistent data structures, and dynamic distance oracles on weighted planar graphs. For the general non-intersecting case, the algorithm achieves a running time of Õ(n^{2−1/48}), significantly improving upon previous results.

Traditional geometric distance queries are non-differentiable, posing a significant barrier to their integration into gradient-based machine learning pipelines. This work proposes a globally differentiable approximation of the Euclidean distance that leverages regularized distances, a partition of unity, and a novel approximate Voronoi diagram. The resulting smooth function achieves a (1+ε)-approximation guarantee while avoiding lifting transformations and ray casting, and exhibits asymptotically optimal bounds on both gradient and Hessian norms. The method enables end-to-end learning and attains state-of-the-art performance in terms of memory overhead and query efficiency for approximate nearest neighbor search, all while preserving theoretically optimal derivative bounds.