A robust and efficient method for solving point distance problems by homotopy

Imbach, Rémi; Mathis, Pascal; Schreck, Pascal

doi:10.1007/s10107-016-1058-7

Cited by 4 publications

(2 citation statements)

References 19 publications

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“…For exhaustive searching of multiple solutions, homotopy continuation was applied (Allgower & Georg, 1993;Durand & Hoffmann, 2000;Imbach et al, 2017;Lamure & Michelucci, 1996). It theoretically guarantees to find all solutions to an equation system.…”

Section: Solvingmentioning

confidence: 99%

A review on geometric constraint solving

Tang¹,

Qiang²,

Feng³

et al. 2022

Preprint

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This paper presents a comprehensive review of geometric constraint solving in parametric computer-aided design (CAD), with the major focus on its advances in the last 15 years. Geometric constraint solving can date back to the very first CAD prototype, Sketchpad, in the 1960s, but serious research studies were carried out only after parametric CAD was introduced in the late 1980s. In the following 30-year history of GCS research, two development stages may be identified: (1) the first 15 years (late 1980s -mid 2000s) were primarily devoted to geometric constraint decomposition for well-constrained systems or those with only structural constraint dependencies; and (2) the second 15 years (late 2000s -now) have seen research efforts shifted towards classification criteria and decomposition algorithms for general constraint systems (with and without non-structural constraint dependencies). Most existing reviews focused on the first 15 years. The problem researched in the second 15 years is, however, equally important, considering that a manually specified constraint system usually contains under-and over-constrained parts, and that such parts must be correctly detected and resolved before numerical solving can work. In this regard, this review paper covers both stages and will discusses what has already been made possible for handling general constraint systems, what developments can be expected in the near future, and which areas remain problematic.

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Section: Solvingmentioning

confidence: 99%

A review on geometric constraint solving

Tang¹,

Qiang²,

Feng³

et al. 2022

Preprint

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“…The other set of methods aim at finding all roots, e.g. homotopy continuation method (Allgower and Georg, 1993;Lamure and Michelucci, 1996;Durand, 1998;Imbach, Mathis, and Schreck, 2017) and subdivision methods (Dokken, 1985;Sherbrooke and Patrikalakis, 1993;Elber and Kim, 2001;Bartoň, 2011). There is always a trade-off between efficiency and exhaustive searching.…”

Section: Geometric Constraint Solvingmentioning

confidence: 99%

Towards computing complete parameter ranges in parametric modeling

Tang¹,

Qiang²,

Gao³

2022

Preprint

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In parametric design, the geometric model is edited by changing relevant parameters in the parametric model, which is commonly done sequentially on multiple parameters. Without guidance on allowable parameter ranges that can guarantee the solvability of the geometric constraint system, the user could assign improper parameter values to the model's parameters, which would further lead to a failure in model updating. However, current commercial CAD systems provide little support for the proper parameter assignments. Although the existing methods can compute allowable ranges for individual parameters, they face difficulties in handling multi-parameter situations. In particular, these methods could miss some feasible parameter values and provide incomplete allowable parameter ranges. To solve this problem, an automatic approach is proposed in this paper to compute complete parameter ranges in multi-parameter editing. In the approach, a set of variable parameters are first selected to be sequentially edited by the user; before each editing operation, the one-dimensional ranges of the variable parameters are presented as guidance. To compute the one-dimensional ranges, each variable parameter is expressed as an equality-constrained function, and its one-dimensional allowable range is obtained by calculating the function range. To effectively obtain the function range which can hardly be calculated in a normal way, the function range problem is converted into a constrained optimization problem, and is then solved by Lagrange multiplier method and the Niching particle swarm optimization algorithm (the NichePSO). The effectiveness and efficiency of the proposed approach is verified by several experimental results.

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