An exact intersection method for triangulated irregular network and its application to 3D geological modeling
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摘要:
针对目前三维地质模型求交方法中精确度不高、数值计算存在误差等问题，提出一种基于不规则三角网、融合浮点精度与精确精度的地质体精确求交算法。首先，构建不规则三角网（TIN） 的方向包围盒（OBB）层次树，通过快速碰撞检测筛选出有效相交三角形对集合，大幅缩减后续计算范围。其次，将相交三角形对分解为边 − 三角形求交单元，建立包含交点空间位置状态（onVertex（顶点上）；onEdge（边上）；inTriangle（面内））及对应实体（点、边、面）
ID 的交点位置关系结构，既消除交点坐标的冗余计算，又通过位置状态与实体ID 匹配精准判别重复交点（如：onVertex 状态下匹配顶点ID ；onEdge 状态下匹配边ID ）。最后，根据拓扑正确性需求动态选择精度模式：若浮点精度可维持拓扑正确，则采用投影降维的约束 Delaunay 三角化（CDT）方法完成重三角化；若出现交点位置偏移（如面内点投影至边外）、点重合或交线段虚假相交等偏差，则切换至有理数表示的精确精度，通过约束点插入（面内点将三角形拆分为3个新三角、边上点将相邻三角形拆分为4个新三角）与约束线段插入（处理相交线段并执行边交换规则：凸四边形直接换边、凹四边形待排队换边）实现重三角化，确保重构结果的拓扑一致性。结果表明，提出的求交算法可以很好地支撑三角网的求交运算，算法转换至精确精度的次数少，具有较好的计算效率及较高的鲁棒性。算法平均时间花销优于GOCAD软件，能够满足大多数不规则三角网复杂地质模型间求交计算的需要。Abstract:To address the problems of low accuracy and numerical errors in the current intersection methods for 3D geological models, this paper proposes a precise intersection algorithm for geological bodies based on the triangulated irregular network (TIN) and integrating floating-point arithmeticwith exact arithmetic. The specific process is as follows: First, a oriented bounding box (OBB) hierarchy for the TIN is constructed, and a set of effectively intersecting triangle pairs is identified through fast collision detection, thereby greatly reducing the scope of subsequent calculations. Second, the intersecting triangle pairs are decomposed into edge-triangle intersection tests, and an intersection positional-relationship data structure is established. This structure includes the spatial position states of intersections (atVertex: at a vertex, onEdge: on an edge, inTriangle: inside the face) and the IDs of corresponding entities (points, edges, faces). It not only eliminates redundant computations of intersection coordinates but also accurately identifies duplicate intersections by matching positional states with entity IDs (e.g., matching vertex IDs in the atVertex state and edge IDs in the onEdge state). Finally, the precision mode is selected dynamically according to the requirements of topological correctness: if floating-point precision can maintain topological correctness, projection-based constrained delaunay triangulation (CDT) is used to perform retriangulation; if deviations such as intersection position offsets (e.g., an inTriangle point projected outside the edge), point coincidence, or spurious intersections between intersection segments occur, the algorithm switches to exact precision represented by rational numbers. Retriangulation is carried out through constrained point insertion (an inTriangle point splits a triangle into three new triangles, and an onEdge point splits adjacent triangles into four new triangles) and constrained line-segment insertion (processing intersecting line segments and implementing edge-swap rules: Direct edge swapping for convex quadrilaterals and queued edge swapping for concave quadrilaterals), thereby ensuring the topological consistency of the reconstructed results. Experimental results show that the proposed algorithm effectively supports intersection operation on triangular networks, requires infrequent conversions to exact precision, and exhibits good computational efficiency and high robustness. The average runtime is lower than that of GOCAD, and the method meets the intersection requirements of most complex geological models based on triangulated irregular networks.
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图 1 三角形对相交
onVertex. 交点位于三角形的顶点上；onEdge. 交点位于三角形的边上；inTriangle. 交点位于三角形内部；T1，T2. 三角形；P1，P2. 交点；下同
Figure 1. Intersection of two triangles
图 2 基于浮点数三角形重构（ABC为三角形；P1～P4均为约束点；下同）
Figure 2. Retriangulating based on floating point number
图 7 地质体地层面间求交
Figure 7. Intersection between stratigraphic planes of a geological body
表 1 实验结果对比
Table 1. Comparison of the experimental results
三角形数/个 求交情况 运算时间/ms 模型1 模型2 相交三
角形/对交点/个 转换/次 本研究
算法GOCAD
软件简单立方体求交 12 12 56 42 0 9 13 地质体地层面间 2182 2 71 71 0 587 2047 断层面与断层面 61712 604 249 251 6 1986 5500 
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