A triangular mesh is the piecewise linear approximation of a sampled or analytical surface, when each patch is a triangle. The connectivity of the mesh can be easily represented using its dual graph. Each node of such a graph has at most three incident edges; if the surface is homeomorphic to a sphere, each node has exactly three incident edges. Several triangular meshes, representing the same surface, with an increasing number of triangles are a representation of the surface at different levels of detail (LOD). When the number of triangles from one LOD to another varies continuously we call such a structure a continuous level of detail (CLOD) approximation of the surface. Given a CLOD data structure we can extract, at each level, the mesh representing the surface and derive its dual graph. If we group the triangles forming each mesh in strips, to accelerate their rendering, we should use two colors for the dual graph's edges to distinguish between the edges linking nodes belonging to the same strip or not. The main goal of this paper is to present a set of rules to recolor the dual graph of the mesh when passing from one LOD to the next and back. The operations used to change the mesh are a Vertex Split (VS) when the resolution increases, and an Edge Collapse (EC) when the resolution decreases. We can, then, use a local topological analysis to derive the rules allowing to recolor the graph, and to show that, under certain conditions, the recoloring is optimal. This allows to keep effectively an optimal triangle strip structure over the mesh, while changing its resolution.

### Rewriting rules for the dual graph of a stripified CLOD mesh

#### Abstract

A triangular mesh is the piecewise linear approximation of a sampled or analytical surface, when each patch is a triangle. The connectivity of the mesh can be easily represented using its dual graph. Each node of such a graph has at most three incident edges; if the surface is homeomorphic to a sphere, each node has exactly three incident edges. Several triangular meshes, representing the same surface, with an increasing number of triangles are a representation of the surface at different levels of detail (LOD). When the number of triangles from one LOD to another varies continuously we call such a structure a continuous level of detail (CLOD) approximation of the surface. Given a CLOD data structure we can extract, at each level, the mesh representing the surface and derive its dual graph. If we group the triangles forming each mesh in strips, to accelerate their rendering, we should use two colors for the dual graph's edges to distinguish between the edges linking nodes belonging to the same strip or not. The main goal of this paper is to present a set of rules to recolor the dual graph of the mesh when passing from one LOD to the next and back. The operations used to change the mesh are a Vertex Split (VS) when the resolution increases, and an Edge Collapse (EC) when the resolution decreases. We can, then, use a local topological analysis to derive the rules allowing to recolor the graph, and to show that, under certain conditions, the recoloring is optimal. This allows to keep effectively an optimal triangle strip structure over the mesh, while changing its resolution.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11584/29003`
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