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Abstract: A novel representation of a triangular mesh surface using a set of scale-invariant measures is proposed. The measures consist of angles of the triangles (triangle angles) and dihedral angles along the edges (edge angles) which are scale and rigidity independent. The vertex coordinates for a mesh give its scale-invariant measures, unique up to scale, rotation, and translation. Based on the representation of mesh using scale-invariant measures, a two-step iterative deformation algorithm is proposed, which can arbitrarily edit the mesh through simple handles interaction. The algorithm can explicitly preserve the local geometric details as much as possible in different scales even under severe editing operations including rotation, scaling, and shearing. The efficiency and robustness of the proposed algorithm are demonstrated by examples.

尺度自动感知的几何体变形技术

研究目的：针对三角形网格的大尺度形变，提出一种基于尺度不变量的变形技术。
研究方法：针对三角形网格中基于顶点--领域的局部微分坐标，提出一套尺度不变的几何量（图1）。基于这套几何不变量，给出尺度自适应的几何体变形能量（方程6）。该复杂方程难以直接求解；为有效求解，利用分离变量原理，设计了一个两步迭代算法（算法1）。将此算法获得的几何变形结果与多种知名的几何变形算法进行比较（图9~11）。最后展示了一系列利用我们的算法进行网格变形的结果（图12~14）。
重要结论：针对三角形网格微分坐标中的尺度不变量，提出了一种新颖的基于尺度不变度量的网格变形技术，使得几何体在大尺度形变过程中能够有效保持几何细节不变。
微分几何坐标；尺度不变的度量；网格变形

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