Re-parameterization reduces irreducible geometric constraint systems

Abstract

International audienceYou recklessly told your boss that solving a non-linear system of size n (n unknowns and n equations) requires a time proportional to n, as you were not very attentive during algorithmic complexity lectures. So now, you have only one night to solve a problem of big size (e.g., 1000 equations/unknowns), otherwise you will be fired in the next morning. The system is well-constrained and structurally irreducible: it does not contain any strictly smaller well-constrained subsystems. Its size is big, so the Newton–Raphson method is too slow and impractical. The most frustrating thing is that if you knew the values of a small number k<<n of key unknowns, then the system would be reducible to small square subsystems and easily solved. You wonder if it would be possible to exploit this reducibility, even without knowing the values of these few key unknowns. This article shows that it is indeed possible. This is done at the lowest level, at the linear algebra routines level, so that numerous solvers (Newton–Raphson, homotopy, and also p-adic methods relying on Hensel lifting) widely involved in geometric constraint solving and CAD applications can benefit from this decomposition with minor modifications. For instance, with k<<n key unknowns, the cost of a Newton iteration becomes O(kn^2) instead of O(n^3). Several experiments showing a significant performance gain of our re-parameterization technique are reported in this paper to consolidate our theoretical findings and to motivate its practical usage for bigger systems

A review on geometric constraint solving

“…Considered that not all geometric constraint systems are ruler-and-compass constructible (Gao & Zhang, 2003), Mathis et al (Mathis et al, 2012) and Barki et al (Barki et al, 2016) proposed a re-parameterization method to make the ruler-and-compass technique available. It is achieved by removing a constraint and adding a new constraint with driving parameter.…”

Exploratory design of mechanical devices with motion constraints

“…Analyzing and solving such constraints have been tackled by Fudos et al [14] in a general setting, and Sitharam et al [31] in the context of mechanisms. Theoretically, even determining parameter bounds characterizing the (constrained) solution domain remains a known difficult topic [4]. Instead, in this work, we seek a general approximation strategy supporting interactive design space exploration for an intuitive workflow.…”

Spirou

“…Analyzing and solving such constraints have been tackled by Fudos et al [1997] in a general se ing, and Sitharam et al [2014] in the context of mechanisms. eoretically, even determining parameter bounds characterizing the (constrained) solution domain remains a known di cult topic [Barki et al 2016]. Instead, in this work, we seek a general approximation strategy that support such interactive design space exploration for an intuitive work ow.…”