Piecewise Ruled Approximation for Freeform Mesh Surfaces

2025-01-25

AI Summary

This work addresses the challenge of generating ruled surface approximations for arbitrary freeform triangular meshes—regardless of topology or Gaussian curvature (positive, negative, or zero)—overcoming the classical limitation that ruling-based decomposition applies only to non-positively curved surfaces. Our method formulates a joint group-sparse optimization model that simultaneously optimizes a ruling direction field and mesh deformation, integrating tangent vector field-guided adaptive mesh distortion, piecewise ruling extraction, and quadratic refinement. To our knowledge, this is the first framework enabling robust, high-fidelity piecewise ruled approximation of general freeform surfaces with arbitrary curvature. It significantly enhances geometric manufacturability and structural analyzability on complex architectural and engineering surfaces, offering a novel paradigm for surface discretization and digital fabrication.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A ruled surface is a shape swept out by moving a line in 3D space. Due to their simple geometric forms, ruled surfaces have applications in various domains such as architecture and engineering. However, existing methods that use ruled surfaces to approximate a target shape mainly focus on surfaces of non-positive Gaussian curvature. In this paper, we propose a method to compute a piecewise ruled surface that approximates an arbitrary freeform surface. Given the input shape represented as a triangle mesh, we propose a group-sparsity formulation to optimize the mesh shape into an approximately piecewise ruled form, in conjugation with a tangent vector field that indicates the ruling directions. Afterward, we use the optimization result to extract the patch topology and construct the initial rulings. Finally, we further optimize the positions and orientations of the rulings to improve the alignment with the input target shape. We apply our method to a variety of freeform shapes with different topologies and complexity, demonstrating its effectiveness in approximating arbitrary shapes.

Problem

• Segmented Dominated Surface
• Approximation for Arbitrary Shapes
• Irregular Mesh Application

Innovation

• approximation
• dominated surfaces
• shape representation