arXiv 1608.052052016-08cs.GRcs.CGcs.DS

树可分解和欠约束几何约束问题

Tree-decomposable and Underconstrained Geometric Constraint Problems

Ioannis Fudos, Christoph M. Hoffmann, Robert Joan-Arinyo

备注：: This work was accepted to appear as a chapter, pending minor revision, to the "Handbook of Geometric Constraints Principles", edited by Meera Sitharam, Audrey St.John and Jessica Sidman, Mathematics series, CRC press (Taylor and Francis group). To appear in 2017. Universitat Politecnica de Catalunya, Research Report: http://hdl.handle.net/2117/91041

摘要：

在本文中，我们关注的是几何约束求解器，即，用程序找出几何约束问题的一个或多个解。如果不存在解，求解器应该宣布没有找到解。由于约束问题的复杂性、类型或难度，即使存在解，求解器也可能找不到解。因此，可能会有假阴性，但永远不会有假阳性。直觉上，找到解决方案的能力可以被认为是衡量解决方案的能力。我们考虑静态约束问题及其求解器。我们不考虑动态约束求解器，也称为动态几何程序，其中特定的几何元素移动，交互式或沿规定的轨迹，同时不断保持所有规定的约束。然而，如果我们有一个解静态约束问题，是足够快和称职的，我们可以建立一个动态的几何程序，通过解决静态问题的运动元素的轨迹足够密集的采样。我们调查的工作有其根源的应用，特别是在机械计算机辅助设计（MCAD）。MCAD中使用的约束求解器在20世纪90年代取得了巨大的飞跃。这些方法通过初始的基于图的结构分析来解决几何约束问题，该结构分析提取通用子问题并确定它们如何组合以形成完整的解决方案。然后，这些子问题被交给代数求解器，该代数求解器解决通用子问题的特定实例并将它们组合起来。

英文摘要：

In this paper, we are concerned with geometric constraint solvers, i.e., with programs that find one or more solutions of a geometric constraint problem. If no solution exists, the solver is expected to announce that no solution has been found. Owing to the complexity, type or difficulty of a constraint problem, it is possible that the solver does not find a solution even though one may exist. Thus, there may be false negatives, but there should never be false positives. Intuitively, the ability to find solutions can be considered a measure of solver's competence. We consider static constraint problems and their solvers. We do not consider dynamic constraint solvers, also known as dynamic geometry programs, in which specific geometric elements are moved, interactively or along prescribed trajectories, while continually maintaining all stipulated constraints. However, if we have a solver for static constraint problems that is sufficiently fast and competent, we can build a dynamic geometry program from it by solving the static problem for a sufficiently dense sampling of the trajectory of the moving element(s). The work we survey has its roots in applications, especially in mechanical computer-aided design (MCAD). The constraint solvers used in MCAD took a quantum leap in the 1990s. These approaches solve a geometric constraint problem by an initial, graph-based structural analysis that extracts generic subproblems and determines how they would combine to form a complete solution. These subproblems are then handed to an algebraic solver that solves the specific instances of the generic subproblems and combines them.

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