同伦法求解点距离问题的一种稳健有效方法
A Robust and Efficient Method for Solving Point Distance Problems by Homotopy
摘要：
点距离解决问题的目标是找到已知一些点对之间距离的点的2D或3D放置。通常的指导方针是通过数值迭代法（\emph{e.g.} Newton-Raphson方法）来求解它们。一个唯一的解决方案，而存在许多。然而，解决方案的数量可以是指数级的，方法应该提供接近用户绘制的草图的解决方案。几何推理可以通过改变几个方程并将其三角化来帮助简化底层方程系统。这种三角化是解决方案的几何构造，称为构造计划。我们的目标是找到几个解决方案接近草图上的一个一维路径定义的全球参数同伦使用建设计划。由于特定的几何构型，可能会遇到一些数值不稳定性。通过实时改变施工计划来解决这一问题，数值结果表明该混合方法是有效的和鲁棒的。
英文摘要：
The goal of Point Distance Solving Problems is to find 2D or 3D placements of points knowing distances between some pairs of points. The common guideline is to solve them by a numerical iterative method (\emph{e.g.} Newton-Raphson method). A sole solution is obtained whereas many exist. However the number of solutions can be exponential and methods should provide solutions close to a sketch drawn by the user.Geometric reasoning can help to simplify the underlying system of equations by changing a few equations and triangularizing it.This triangularization is a geometric construction of solutions, called construction plan. We aim at finding several solutions close to the sketch on a one-dimensional path defined by a global parameter-homotopy using a construction plan. Some numerical instabilities may be encountered due to specific geometric configurations. We address this problem by changing on-the-fly the construction plan.Numerical results show that this hybrid method is efficient and robust.
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