C. Gotsman and V. Surazhsky, "Guaranteed intersection-free polygon morphing," Computers and Graphics, vol. 25, no. 1, pp. 67--75, 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Gert Vegter z Abstract This paper provides a short and intuitive proof of Tutte s barycentric embedding theorem [14] compared to the original one which involved a lot of graph theory and complicated terminology. We then study a method closely related to papers by Gotsman, Surazhsky and Floater [8, 9, 10] to build isotopies of triangulations in the plane, whose theoretical basis is Tutte s theorem. Our method works on a smaller class of embeddings, but has a clear physical interpretation. We nally show that Tutte s result does not extend to 3D space, reviewing also a paper by Starbird [12] 1 ....

....Colin de Verdi ere [5] shows the result on arbitrary surfaces of non positive curvature using the Gauss Bonnet formula, but his proof is valid only for triangulated graphs and does not seem to extend to 3 connected graphs. Isotopies. Tutte s theorem yields a method, described by Gotsman et al. in [8, 9, 10], to morph two triangulations, the boundary being the same convex polygon in both embeddings. One can compute coecients uv 0 for each interior vertex u and each neighbor v of u so that u is the barycenter with coecients ( uv ) v of its neighbors in the initial embedding. Doing the same for ....

C. Gotsman and V. Surazhsky. Guaranteed intersection-free polygon morphing. Comput. and Graphics, 25(1):67-75, 2001.

.... can be done on the cubic grid of size 2n 169n 3 [27] For a survey on algorithms for graph drawing, see [14] Tutte s method is often used in graphics applications related to surface parametrization in multiresolution problems [16] and geometric modeling [19] texture mapping [25] and morphing [23, 20, 21]. In his paper [38] in addition to showing Theorem 1, Tutte simultaneously proves again Kuratowski s planarity criterion [24] of 1930: a graph is planar unless it contains a subdivision of one of the two Kuratowski graphs K 5 and K 3;3 . The proofs of both results are entangled together in ....

....to a face are not on the same line. This step uses the nonplanarity of K 3;3 together with simple geometric ideas. Then, the generalization to arbitrary 3connected graphs comes easily. Isotopies. Tutte s theorem yields a method, described by Floater and Gotsman [20] and Gotsman and Surazhsky [21], to morph two triangulations, the boundary being the same convex polygon in both embeddings. One can compute coecients uv 0, for each interior vertex u and each neighbor v of u, so that u is the barycenter with coe cients ( uv ) v of its neighbors in the initial embedding. Doing the same ....

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C. Gotsman and V. Surazhsky. Guaranteed intersection-free polygon morphing. Computers and Graphics, 25(1):67-75, 2001.

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C. Gotsman and V. Surazhsky, "Guaranteed intersection-free polygon morphing," Computers and Graphics, vol. 25, no. 1, pp. 67--75, 2001.

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C.Gotsman and V.Surazhsky, Guaranteed Intersection-Free Polygon Morphing. Computers and Graphics, 25(1):67-75, 2001.

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GOTSMAN, C., AND SURAHHSKY, V. Guaranteed Intersection-free Polygon Morphing. Computer and Graphics 25, 1 (2001), pp.67--75.

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