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Robust Polylines Tracing for N-Symmetry Direction Field on Triangulated Surfaces

机译:三角表面上N对称方向场的鲁棒折线追踪

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We are proposing an algorithm for tracing polylines that are oriented by a direction field defined on a triangle mesh. The challenge is to ensure that two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is impossible to enforce with existing algorithms. The core of our contribution is to determine how polylines cross each triangle. Our solution is inspired by EdgeMaps where each triangle boundary is decomposed into inflow and outflow intervals such that each inflow interval is mapped onto an outflow interval. To cross a triangle, we find the inflow interval that contains the entry point, and link it to the corresponding outflow interval, with the same barycentric coordinate. To ensure that polylines cannot merge or cross, we introduce a new direction field representation, we resolve the inflow/outflow interval pairing with a guaranteed combinatorial algorithm, and propagate the barycentric positions with arbitrary precision number representation. Using these techniques, two streamlines crossing the same triangle cannot merge or cross, but only locally overlap when all streamline extremities are located on the same edge. Cross-free and merge-free polylines can be traced on the mesh by it-eratively crossing triangles. Vector field singularities and polyline/vertex crossing are characterized and consistently handled.
机译:我们正在提出一种算法,用于跟踪由三角形网格上定义的方向场定向的折线。挑战在于确保两条这样的折线不能交叉或合并。此属性是网格分割的基础,无法用现有算法强制执行。我们所做贡献的核心是确定折线如何与每个三角形交叉。我们的解决方案受到EdgeMaps的启发,其中每个三角形边界都分解为流入和流出间隔,因此每个流入间隔都映射到流出间隔上。为了穿过三角形,我们找到包含入口点的流入间隔,并将其链接到具有相同重心坐标的相应流出间隔。为了确保折线不能合并或交叉,我们引入了新的方向场表示,我们使用有保证的组合算法解决了流入/流出间隔配对,并使用任意精度的数字表示来传播重心位置。使用这些技术,穿过同一三角形的两条流线不能合并或交叉,而当所有流线末端位于同一边缘时,它们只能局部重叠。可以通过交叉交叉的三角形在网格上跟踪自由交叉和自由合并的折线。矢量场奇异性和折线/顶点交点具有特征性并得到一致处理。

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