网格属性 Mesh properties

• 流型属性manifold property

  • 边流型 edge_manifold：一条边只能属于一个或者两个三角形
  • 顶点流型 vertex_manifold： 若顶点的星形结构同时满足边流形且边连通（例如两个或多个面仅通过顶点相连而非边相连），则该三角网格的顶点为流形。
    • 边流通：在顶点流形的定义中，边连通指的是：围绕一个顶点的所有三角形面片必须通过边（edges）连接成一个连续的“扇形”或“星形”结构。

• 自相交 self intersecting：exists a triangle in the mesh that is intersecting another mesh

• 水密 watertight

  • edge manifold
  • vertex manifold
  • not self intersecting

• orientable
  对于一个三角网格，“可定向” 意味着：你可以为网格中的每一个三角形分配一个环绕顺序（例如，从其正面看，顶点是逆时针排列的 (v0, v1, v2)），使得任何两个共享一条边的三角形，在这条公共边上诱导出的顶点顺序恰好相反。
  或者说所有的法线朝向网格外

  • 可定向曲面：像一个普通的圆柱体侧面。你可以用一致的箭头方向（比如总是顺时针）画出所有三角形的边界，并且相邻三角形的共享边上的箭头方向总是相反的（一个三角形的出边是另一个的入边）。你可以清晰地定义“内壁”和“外壁”。
  • 不可定向曲面：像一个莫比乌斯环。如果你尝试用同样的方法去一致地定义所有三角形的方向，你会发现当你沿着环走一圈回到起点时，你最初定义为“正面”的面，莫名其妙地变成了“背面”。它只有一个面和一条边界，你无法区分“内”和“外”。

Mesh - Open3D 0.19.0 documentation

A triangle mesh has several properties that can be tested with Open3D. One important property is the manifold property, where we can test the triangle mesh if it is edge manifold is_edge_manifold and if it is is_vertex_manifold. A triangle mesh is edge manifold, if each edge is bounding either one or two triangles. The function is_edge_manifold has the bool parameter allow_boundary_edges that defines if boundary edges should be allowed. Further, a triangle mesh is vertex manifold if the star of the vertex is edge-manifold and edge-connected, e.g., two or more faces connected only by a vertex and not by an edge.

Another property is the test of self-intersection. The function is_self_intersecting returns True if there exists a triangle in the mesh that is intersecting another mesh. A watertight mesh can be defined as a mesh that is edge manifold, vertex manifold and not self intersecting. The function is_watertight implements this check in Open3D.

We also can test the triangle mesh, if it is orientable, i.e. the triangles can be oriented in such a way that all normals point towards the outside. The corresponding function in Open3D is called is_orientable.

The code below tests a number of triangle meshes against those properties and visualizes the results. Non-manifold edges are shown in red, boundary edges in green, non-manifold vertices are visualized as green points, and self-intersecting triangles are shown in pink.

A watertight mesh can be defined as a mesh that is edge manifold, vertex manifold and not self intersecting.