Solving Euclidean Problems by Isotropic Initialization

2025-06-02

AI Summary

Solving Euclidean geometric constraint systems in computational design and manufacturing remains challenging due to poor convergence, low robustness, and high computational cost. Method: This paper introduces a novel initialization paradigm bridging Euclidean and isotropic geometry: constraints are first solved efficiently in an isotropic space—preserving structural properties and enabling analytic tractability—then the solution is mapped back to Euclidean space to guide gradient-based optimization. Contribution/Results: We establish the first systematic mapping framework between Euclidean and isotropic geometries, enabling geometry-aware initial solution generation. Integrating isotropic modeling, constraint projection, mesh parameterization, and joint discrete asymptotic/geodesic curve representation, our method successfully solves three complex mesh structures: quadrilateral mechanism meshes, composite asymptotic–geodesic shells, and constant-node-angle asymptotic gridshells. Experiments demonstrate significantly improved convergence speed and solution stability compared to conventional Euclidean-only approaches.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Many problems in Euclidean geometry, arising in computational design and fabrication, amount to a system of constraints, which is challenging to solve. We suggest a new general approach to the solution, which is to start with analogous problems in isotropic geometry. Isotropic geometry can be viewed as a structure-preserving simplification of Euclidean geometry. The solutions found in the isotropic case give insight and can initialize optimization algorithms to solve the original Euclidean problems. We illustrate this general approach with three examples: quad-mesh mechanisms, composite asymptotic-geodesic gridshells, and asymptotic gridshells with constant node angle.

Problem

• Solving Euclidean geometry constraints in design
• Using isotropic geometry to simplify problems
• Initializing optimization via isotropic solutions

Innovation

• Using isotropic geometry for initialization
• Applying structure-preserving Euclidean simplification
• Optimizing solutions via isotropic insights