This might require linearly searching through the

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or edge with a given triangle. This might require linearly searching through the triangle list to determine whether they share a vertex or two. If there are millions
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62 S. M. LaValle: Virtual Reality Figure 3.3: Part of a doubly connected edge list is shown here for a face that has five edges on its boundary. Each half-edge structure e stores pointers to the next and previous edges along the face boundary. It also stores a pointer to its twin half-edge, which is part of the boundary of the adjacent face. (Figure from Wikipedia user Accountalive). of triangles, which is not uncommon, then it would cost too much to perform this operation repeatedly. For these reasons and more, geometric models are usually encoded in clever data structures. The choice of the data structure should depend on which opera- tions will be performed on the model. One of the most useful and common is the doubly connected edge list , also known as half-edge data structure [20, 63]. See Fig- ure 3.3. In this and similar data structures, there are three kinds of data elements: faces , edges , and vertices . These represent two, one, and zero-dimensional parts, respectively, of the model. In our case, every face element represents a triangle. Each edge represents the border of one or two, without duplication. Each vertex is shared between one or more triangles, again without duplication. The data structure contains pointers between adjacent faces, edges, and vertices so that al- gorithms can quickly traverse the model components in a way that corresponds to how they are connected together. Inside vs. outside Now consider the question of whether the object interior is part of the model (recall Figure 3.2). Suppose the mesh triangles fit together perfectly so that every edge borders exactly two triangles and no triangles intersect unless they are adjacent along the surface. In this case, the model forms a complete barrier between the inside and outside of the object. If we were to hypothetically fill the inside with a gas, then it could not leak to the outside. This is an example of a coherent model . Such models are required if the notion of inside or outside is critical to the VWG. For example, a penny could be inside of the dolphin, but not intersecting with any of its boundary triangles. Would this ever need to be detected? If we remove a single triangle, then the hypothetical gas would leak
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3.1. GEOMETRIC MODELS 63 out. There would no longer be a clear distinction between the inside and outside of the object, making it di cult to answer the question about the penny and the dolphin. In the extreme case, we could have a single triangle in space. There is clearly no natural inside or outside. Furthermore, the model could be as bad as polygon soup , which is a jumble of triangles that do not fit together nicely and could even have intersecting interiors. In conclusion, be careful when constructing models so that the operations you want to perform later will be logically clear.
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Christopher Reinemann
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