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![5: Adjacent double tetrahedra, coeecient signs and C 1 continuity relations for two cubic A-patches deened on non-convex adjacent faces any polyhedron P and yield a globally C 1 surface. This scheme is also extended to a C 2 version using A-patches of degree ve 8]. Details are provided in the next subsection. In these algorithms 9, 7] they rst specify unique \normals" (tangent planes) on the vertices of P, then build a simplicial hull surrounding the surface triangulation T of P and satisfying vertex tangent plane containment, and nally construct cubic A-patches within each tetrahedron of the simplicial hull. Diierent conngurations of vertex \normals" for edges and faces of T are categorized as`convexas`convex' and`non and`non-convex'. The edges and faces together with their normals are thus tagged as`convexas`convex' and`nonand`nonconvex'. As part of the simplicial hull, a single tetrahedron is constructed for a `convex' face while a pair of tetrahedra are constructed (one on each side of the face) for a `non-convex' face (4). C 1 continuity conditions are next set up and satisifed between coeecients (control points) of all adjacent tetrahedra in the simplicial hull. 5 shows the control points, their signs and their relations (like numbers) by C 1 continuity conditions in neighboring edge and face tetrahedra for the most diicult of cases, viz. two adjacent`nonadjacent`non-convex' faces. The C 1 continuity conditions are all linear and shown to be always satissable while maintaining the single sign change conditions of the A-patches. Details are in 9]. The resulting mesh of A-patches is thus guaranteed to be globally C 1 continuous. They also show how to adjust the free parameters of the A-patches to achieve both local and global shape control (bottom of 6). They use a single cubic A-patch per face of T except for the following two special cases. For a `non-convex' face, if additionally the three inner products of the face normal and its three adjacent face normals have diierent signs, then in this case one needs to subdivide the face using a single face Clough-Tocher split, yielding C 1 continuity with the help of three cubic A-patches for that face. Furthermore for coplanar adjacent faces of T , they show that the C 1 conditions cannot be met using a single cubic A-patch for each face. Hence for this case they again use face Clough-Tocher splits for the pair of coplanar faces yielding C 1 continuity with the help of three cubic A-patches per face.](profile/Chandrajit-Bajaj/publication/2388672/figure/fig3/AS:669491453231114@1536630609447/Adjacent-double-tetrahedra-coeecient-signs-and-C-1-continuity-relations-for-two-cubic.png)
5: Adjacent double tetrahedra, coeecient signs and C 1 continuity relations for two cubic A-patches deened on non-convex adjacent faces any polyhedron P and yield a globally C 1 surface. This scheme is also extended to a C 2 version using A-patches of degree ve 8]. Details are provided in the next subsection. In these algorithms 9, 7] they rst specify unique \normals" (tangent planes) on the vertices of P, then build a simplicial hull surrounding the surface triangulation T of P and satisfying vertex tangent plane containment, and nally construct cubic A-patches within each tetrahedron of the simplicial hull. Diierent conngurations of vertex \normals" for edges and faces of T are categorized as`convexas`convex' and`non and`non-convex'. The edges and faces together with their normals are thus tagged as`convexas`convex' and`nonand`nonconvex'. As part of the simplicial hull, a single tetrahedron is constructed for a `convex' face while a pair of tetrahedra are constructed (one on each side of the face) for a `non-convex' face (4). C 1 continuity conditions are next set up and satisifed between coeecients (control points) of all adjacent tetrahedra in the simplicial hull. 5 shows the control points, their signs and their relations (like numbers) by C 1 continuity conditions in neighboring edge and face tetrahedra for the most diicult of cases, viz. two adjacent`nonadjacent`non-convex' faces. The C 1 continuity conditions are all linear and shown to be always satissable while maintaining the single sign change conditions of the A-patches. Details are in 9]. The resulting mesh of A-patches is thus guaranteed to be globally C 1 continuous. They also show how to adjust the free parameters of the A-patches to achieve both local and global shape control (bottom of 6). They use a single cubic A-patch per face of T except for the following two special cases. For a `non-convex' face, if additionally the three inner products of the face normal and its three adjacent face normals have diierent signs, then in this case one needs to subdivide the face using a single face Clough-Tocher split, yielding C 1 continuity with the help of three cubic A-patches for that face. Furthermore for coplanar adjacent faces of T , they show that the C 1 conditions cannot be met using a single cubic A-patch for each face. Hence for this case they again use face Clough-Tocher splits for the pair of coplanar faces yielding C 1 continuity with the help of three cubic A-patches per face.
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Introduction While it is possible to model a general closed surface of arbitrary genus as a single implicit surface patch, the geometry of such a global surface is difficult to specify, interactively control, and polygonize. The main difficulties stem from the fact that implicit representations are iso-contours which generally have multiple real sh...
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