Solving Euclidean Problems by Isotropic Initialization

TL;DR

This paper introduces a novel approach to solving Euclidean geometry problems by leveraging isotropic geometry, simplifying complex constraints and improving initialization for optimization algorithms in design and fabrication tasks.

Contribution

The paper proposes using isotropic geometry as a structure-preserving simplification to aid in solving Euclidean problems, providing a new general methodology with practical examples.

Findings

Isotropic solutions provide valuable insights for Euclidean problems.

Initialization with isotropic solutions improves optimization convergence.

Applicable to complex geometric design problems.

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.