I am reading an article that describes how to represent land and seafloor surfaces with a bunch of triangles joined at their edges.
The paper says this:
As employed by ocean modelers, a mesh is the product of a triangulation of a set of points in the plane, so it should therefore satisfy Euler's formula for a simple polygon, whose characteristic equals one. By convention, such polygons are defined in terms of undirected graphs of vertices, edges, and faces that are, in our case, triangular. Thus, we require that V - E + T = 1, where V = vertices, E (undirected) edges, and T triangles appear in a mesh.
What is the practical implication of a mesh of triangles satisfying Euler's formula? Does it mean that no triangle will overlap another triangle (see Figure 1 below)? Or, does it mean that no triangle will touch another triangle in the middle of an edge (see Figure 2 below)? Or, does it mean something else?
