The vast majority of mesh-based modelling applications iteratively transform the mesh vertices under prescribed geometric conditions. This occurs in particular in methods cycling through the constraint set such as Position-Based Dynamics (PBD). A common case is the approximate local area preservation of triangular 2D meshes under external editing constraints. At the constraint level, this yields the nonconvex optimal triangle projection under prescribed area problem, for which there does not currently exist a direct solution method. In current PBD implementations, the area preservation constraint is linearised. The solution comes out through the iterations, without a guarantee of optimality, and the process may fail for degenerate inputs where the vertices are colinear or colocated. We propose a closed-form solution method and its numerically robust algebraic implementation. Our method handles degenerate inputs through a two-case analysis of the problem's generic ambiguities. We show in a series of experiments in area-based 2D mesh editing that using optimal projection in place of area constraint linearisation in PBD speeds up and stabilises convergence.

中文翻译：

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具有指定面积和方向的最佳三角形投影仪，在基于位置的动力学中的应用

绝大多数基于网格的建模应用程序在指定的几何条件下迭代地变换网格顶点。这尤其发生在循环通过约束集的方法中，例如基于位置的动力学（PBD）。一个常见的情况是在外部编辑约束下三角形2D网格的近似局部保留。在约束级别，这将产生在规定面积问题下的非凸最优三角投影，目前尚不存在直接求解方法。在当前的PBD实现中，区域保留约束是线性化的。解决方案是通过迭代得出的，而没有保证最优性，并且对于顶点共线或共置的退化输入，该过程可能会失败。我们提出了一种封闭形式的求解方法及其数值鲁棒的代数实现。我们的方法通过对问题的通用歧义的两种情况分析来处理退化的输入。我们在基于区域的2D网格编辑中进行的一系列实验表明，在PBD中使用最佳投影代替区域约束线性化可以加快并稳定收敛。