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A natural metric in 2-manifold surfaces is to use geodesic distance. If a 2-manifold surface is represented by a triangle mesh T , the geodesic metric on T can be computed exactly using computational geometry methods. Previous work for establishing the geodesic metric on T only supports using half-edge data structures; i.e., each edge e in T is split into two halves (he1, he2) and each half-edge corresponds to one of two faces incident to e. In this paper, we prove that the exact-geodesic structures on two halfedges of e can be merged into one structure associated with e. Four merits are achieved based on the properties which are studied in this paper: (1) Existing CAD systems that use edge-based data structures can directly add the geodesic distance function without changing the kernel to a half-edge data structure; (2) To find the geodesic path from inquiry points to the source, the MMP algorithm can be run in an onthe- fly fashion such that the inquiry points are covered by correct wedges; (3) The MMP algorithm is sped up by pruning unnecessary wedges during the wedge propagation process; (4) The storage of the MMP algorithm is reduced since fewer wedges need to be stored in an edge-based data structure. Experimental results show that when compared to the classic half-edge data structure, the edge-based implementation of the MMP algorithm reduces running time by 44% and storage by 29% on average.

No abstract uploaded!

No abstract uploaded!

We consider singular integral operators of the form (a)$Z$_{1}L^{−1}Z_{2}, (b)$Z$_{1}Z_{2}L^{−1}, and (c)$L$^{−1}Z_{1}Z_{2}, where$Z$_{1}and$Z$_{2}are nonzero right-invariant vector fields, and$L$is the$L$^{2}-closure of a canonical Laplacian. The operators (a) are shown to be bounded on$L$^{p}for all$p$∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type ($p, p$) for any$p$∈[1, ∞).