W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....patches (RBB) or triangular rational B Splines. The adaptive triangulation is additionally useful for a rapid display and animation of the implicit surface. 1 Introduction Many authors use real algebraic surfaces to cope with the problem of modeling complicated shapes [6] 5] 11] 13] [15], 16] 17] 26] 29] 30] 33] 34] 35] Implicitly defined algebraic surfaces have both advantages, and disadvantages over functional and parametric surfaces[4] The class of implicit algebraic surfaces is closed under several geometric operations (intersections, union, offset, etc. ....

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, 1989.

....of the constructed smooth surface at the vertex. b. Build a surrounding simplicial hull (consisting of a series of tetrahedra) of the triangulation. c. Construct a piecewise trivariate polynomial F within that simplicial hull, and use the zero contour of F to represent the surface. Dahmen [5] rst Project 19671081 supported partially by NSFC. Supported in part by NSF grants ACI 9982297 and KDI DMS 9873326 (d) e) f) Fig 1.1: a) Input mesh consists of three, four and ve sided polygons; b) Constructed hull; c) A zoom in four sided patch; d) A zoom in ve sided patch; e) ....

....the vertices and have no self intersection. Since such simplicial hulls are nontrivial to construct for arbitrary triangulation, several improvements have been made in later publications to overcome the diculties (see [3] 6] 9] 10] For the construction of the surface within , Dahmen [5] used six quadric patches for each face tetrahedron and four quadric patches for each edge tetrahedron. Guo [9] uses a Clough Tocher split to subdivide each face tetrahedron of the simplicial hull, hence utilizing six cubic patches per face of L. The edge tetrahedra are subdivided into two. Dahmen ....

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181-193. Academic Press, Boston, 1989.

....of the implementation of these algorithms and approximation error bounds are also provided. 1 Introduction In CAGD and solid modeling, implicitly defined algebraic surfaces have recently become more and more popular. Many authors use them to cope with the problem of geometric modeling [6] [12], 19] 20] 21] 22] 5] Implicitly defined algebraic surfaces have specific advantages over the functional and parametric surfaces. Among them, the most important ones are: the class of algebraic surfaces is closed under several geometric operations (intersections, union, offset, etc. ....

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, 1989.

....the normal of the constructed smooth surface at the vertex. b. Build a surrounding simplicial hull P #consisting of a series of tetrahedra# of the triangulation. c. Construct a piecewise trivariate polynomial F within that simplicial hull, and use the zero contour of F to represent the surface. Dahmen #Dahmen, 1989# #rst proposed an approach for constructing a simplicial hull of T . In this approach, for each face #p i p j p k #ofT ,two points u ijk and v ijk o# each side of the face are chosen and two tetrahedra #p i p j p k u ijk # and #p i p j p k u ijk ##called face tetrahedra# are constructed. For each ....

....such simplicial hulls are nontrivial to construct for arbitrary triangulation, several improvements have been made in later publications to overcome the di#culties#see Dahmen and Thamm Schaar, 1993; Guo, 1991a; Guo, 1991b; Bajaj et al. 1995b#. For the construction of the surface within P , Dahmen #Dahmen, 1989# used six quadric patches for each face tetrahedron and four quadric patches for each edge tetrahedron. Guo#Guo, 1991a# uses a Clough Tocher split to subdivide each face tetrahedron of the simplicial hull, hence utilizing three cubic patches per face of T . The edge tetrahedra are subdivided into ....

Dahmen, W. #1989#. Smooth piecewise quadratic surfaces. In Lyche, T. and Schumaker, L., editors, Mathematical Methods in Computer AidedGeometric Design, pages 181#193. Academic Press, Boston.

.... M has attracted the interest of many authors (for a review, see [11] Several papers focus on building a piecewise linear surface [12, 13, 18, 25, 36] Other methods use parametric or functional surface patches for either local or global interpolation [9, 22, 24, 27, 32, 34] A few papers (see [3, 14, 15, 23, 28, 35]) use implicit surface patches. In this paper, we use an implicitly defined tensor product polynomial spline surface to approximate the unknown surface M. The problem of interpolating or approximating data defined over a given manifold in R 3 is commonly referred to as modeling 3D scattered ....

Dahmen, W. Smooth piecewise quadratic surfaces. In Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. Schumaker, Eds. Academic Press, Boston, 1989, pp. 181--193.

....of generating A patches. The second problem we consider is how to join a collection of A patches to form a C 1 smooth surface interpolating scattered data points and respecting the topology of a given surface triangulation T of the points. For this problem, prior approaches have been given by [Dah89] using quadric patches, DTS93, Guo91a, Guo91b] using cubic patches and [BI92] using quintic for convex triangulations and degree seven patches for arbitrary surface triangulations T . All these papers provide heuristics to overcome the multiple sheeted and singularity problems of implicit ....

....to overcome the multiple sheeted and singularity problems of implicit patches. In this paper our cubic A patches are guaranteed to be nonsingular and single sheeted within each tetrahedron. While the details of the methods of [DTS93] and [Guo91b] differ somewhat, they both use the scheme of [Dah89] of building a surrounding simplicial hull (consisting of a series of tetrahedra) of the given triangulation T . Such a simplicial hull is nontrivial to construct for triangulations and neither of the papers [Dah89, DTS93, Guo91a, Guo91b] enumerate the different exceptional cases (possible even ....

[Article contains additional citation context not shown here]

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, 1989.

....patch of a trivariate polynomial in Bernstein B ezier(BB) form defined within a tetrahedron[BCX94] where A stands for algebraic. Solutions to the problem of constructing a C 1 mesh of implicit algebraic patches which interpolate the vertices of a simplicial polyhedron P have been given by [Dah89] using quadric patches, BCX94, DTS93, Guo91b, Guo93] using cubic patches and [BI92b] using quintic for convex P(all faces are triangular) and degree seven patches for arbitrary P . While papers [BI92b, Dah89, DTS93, Guo91b, Guo93] provide heuristics based on monotonicity and least square ....

....patches which interpolate the vertices of a simplicial polyhedron P have been given by [Dah89] using quadric patches, BCX94, DTS93, Guo91b, Guo93] using cubic patches and [BI92b] using quintic for convex P(all faces are triangular) and degree seven patches for arbitrary P . While papers [BI92b, Dah89, DTS93, Guo91b, Guo93] provide heuristics based on monotonicity and least square approximation to circumvent the multiple sheeted and singularity problems of implicit patches, BCX94] introduces new sufficiency conditions for the BB form of trivariate polynomials within a tetrahedron, such that ....

[Article contains additional citation context not shown here]

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, 1989.

....the normal of the constructed smooth surface at the vertex. b. Build a surrounding simplicial hull P (consisting of a series of tetrahedra) of the triangulation. c. Construct a piecewise trivariate polynomial F within that simplicial hull, and use the zero contour of F to represent the surface. Dahmen (Dahmen, 1989) first proposed an approach for constructing a simplicial hull of T . In this approach, for each face [p i p j p k ] of T , two points u ijk and v ijk off each side of the face are chosen and two tetrahedra [p i p j p k u ijk ] and [p i p j p k u ijk ] called face tetrahedra) are constructed. For ....

.... such simplicial hulls are nontrivial to construct for arbitrary triangulation, several improvements have been made in later publications to overcome the difficulties(see Dahmen and Thamm Schaar, 1993; Guo, 1991a; Guo, 1991b; Bajaj et al. 1995b) For the construction of the surface within P , Dahmen (Dahmen, 1989) used six quadric patches for each face tetrahedron and four quadric patches for each edge tetrahedron. Guo(Guo, 1991a) uses a Clough Tocher split to subdivide each face tetrahedron of the simplicial hull, hence utilizing three cubic patches per face of T . The edge tetrahedra are subdivided into ....

Dahmen, W. (1989). Smooth piecewise quadratic surfaces. In Lyche, T. and Schumaker, L., editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston.

....condition is difficult to satisfy in general, and does not in general avoid singularities on the zero contour. Another problem is how to stitch together a collection of patches so that they join with the desired derivative continuity. For this problem, partial solutions have been given by Dahmen [18] using quadric patches, Dahmen and Thamm Schaar [19] Lodha [35] and Guo [26, 27] using cubic patches and Bajaj and Ihm [6] using quintic for convex triangulations and degree seven patches for arbitrary surface triangulations. All these papers provide heuristics to overcome the multiple sheeted ....

....Bajaj and Ihm [6] using quintic for convex triangulations and degree seven patches for arbitrary surface triangulations. All these papers provide heuristics to overcome the multiple sheeted and singularity problems of implicit patches. All these methods use variations of the scheme described in [18] of building a surrounding simplicial hull of a given triangulation. Such a construction is nontrivial and none of the papers cited enumerate the exceptional cases (possible even for convex triangulations) nor provide solutions to overcome them. Related papers on approximating scattered data using ....

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, 1989.

....M has attracted the interest of many authors. A number of methods have been developed for its solution [14, 5, 28, 9, 26, 7, 23, 20, 30, 17, 27, 8, 22, 2] Most of the known methods use parametric or functional surface patches in either local interpolation or global interpolation. A few papers (see [29, 10, 18, 4, 11, 3, 2]) use implicit surface patches. In this paper, we use a piecewise implicitly defined tensor product algebraic surface to approximate the unknown surface M . The problem of interpolating data defined over a given manifold in R 3 is commonly referred to as modeling 3D scattered manifold data or ....

DAHMEN, W. Smooth piecewise quadratic surfaces. In Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. Schumaker, Eds. Academic Press, Boston, 1989, pp. 181--193.

....patches (RBB) or triangular rational B Splines. The adaptive triangulation is additionally useful for a rapid display and animation of the implicit surface. 1 Introduction Many authors use real algebraic surfaces to cope with the problem of modeling complicated shapes [6] 5] 11] 13] [15], 16] 17] 26] 29] 30] 33] 34] 35] Implicitly defined algebraic surfaces have both advantages, and disadvantages over functional and parametric surfaces[4] The class of implicit algebraic surfaces is closed under several geometric operations (intersections, union, offset, etc. ....

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, 1989.

....need a complete mesh covering the entire space. Furthermore, a vertex does not need to be fully covered by the trihedral corners of the incident tetrahedra. An incomplete vertex helps introduce degree bounded vertices as well. A simplicial hull construction of this kind first appears in Dahmen [Dah89] and used by Guo and Dahmen and Thamm Scharr. This simplicial hull was subsequently developed to handle cases of arbitrary polyhedra in Bajaj, Chen and Xu[BCX95b, DTS93, Guo91, Lod92] Given a triangulated polyhedron, the simplicial hull construction builds a tetrahedron on each triangular face ....

....tetrahedra. The bottom left picture shows four A patches, one per tetrahedron, joining C 1 smoothly. The bottom right picture shows the smoothed icosahedron where indvidual patches are shaded differently. B1: A Patches 12 4. 1 Smooth Interpolation of a Polyhedron with C 1 A patches Dahmen [Dah89] presents a simplicial hull scheme for constructing C 1 continuous piecewise quadric surface patches for a triangulation T of a polyhedron P . In his simplicial hull construction, each triangular face is covered by a tetrahedron. Similar to the Powell Sabin split [PS77] the tetrahedron is ....

[Article contains additional citation context not shown here]

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, Massachusetts, 1989.

....need a complete mesh covering the entire space. Furthermore, a vertex does not need to be fully covered by the trihedral corners of the incident tetrahedra. An incomplete vertex helps introduce degree bounded vertices as well. A simplicial hull construction of this kind first appears in Dahmen [Dah89] and used by Guo and Dahmen and Thamm Scharr. This simplicial hull was subsequently developed to handle cases of arbitrary polyhedra in Bajaj, Chen and Xu[BCX95b, DTS93, Guo91, Lod92] Given a triangulated polyhedron, the simplicial hull construction builds a tetrahedron on each triangular face ....

....if the surface does not have to pass through any hull vertex, surface modification can be done solely by changing the coefficients of the polynomial. This fact is exploited by the approximating patch scheme of section 4:2. 4. 1 Smooth Interpolation of a Polyhedron with C 1 A patches Dahmen [Dah89] presents a simplicial hull scheme for constructing C 1 continuous piecewise quadric surface patches for a triangulation T of a polyhedron P . In his simplicial hull construction, each triangular face is covered by a tetrahedron. Similar to the Powell Sabin split [PS77] the tetrahedron is ....

[Article contains additional citation context not shown here]

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, Massachusetts, 1989.

....mesh covering the entire space. Furthermore, a vertex does not need to be fully covered by the trihedral corners of the incident tetrahedra. A incomplete vertex helps introduce degree bounded vertices as well. An effective simplicial hull construction of this kind first appears in Dahmen [19], and subsequently developed and used by Guo, Lodha, Dahmen and Thamm Scharr, Bajaj, Chen and Xu[9, 20, 24, 29] Given a triangulated polyhedron, the simplicial hull construction builds a tetrahedron on each triangular face (sometimes a pair of tetrahedra one on each side of a face) See Figure ....

....involve both changes of the coefficients and the simplicial hull, whereas if the surface does not have to pass through any hull vertex, surface modification can be done solely by changing the coefficients of the polynomial. 3. 1 Smooth Interpolation of a Polyhedron with C 1 A patches Dahmen [19] presents a simplicial hull scheme for constructing C 1 continuous piecewise quadric surface patches for a triangulation T of a polyhedron P . In his simplicial hull construction, each triangular face is covered by a tetrahedron. Similar to the Powell Sabin split [40] the tetrahedron is then ....

[Article contains additional citation context not shown here]

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, Massachusetts, 1989.

....there are currently two main approaches to defining the individual pieces of a function whose zero set represents a surface. The first is to generate an approximate triangulation of the surface and then erect a shell like structure of trivariate polynomial pieces over this parametrization [20] [5], 9] 4] 1] 14] The second is to define a function on a regular, global lattice, for example, a piecewise triquadratic, C 1 tensor product spline [15] The regular lattice has the advantage that non rectilinear features of the surface do not require special treatment and that no ....

Dahmen, W. Smooth piecewise quadratic surfaces. In Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. Schumaker, Eds. Academic Press, Boston, 1989, pp. 181--193.

....and single sheeted zero contour patch of a trivariate polynomial in Bernstein B ezier(BB) form defined within a tetrahedron[BCX94a] where the A stands for algebraic. Solutions to the problem of constructing a C 1 mesh of implicit algebraic patches based on polyhedron P have been given by [Dah89, BCX94a, BCX94b, DTS93, Guo91, Guo93, BI92]. While papers [BI92, Dah89, DTS93, Guo91, Guo93] provide heuristics based on monotonicity and least square approximation to circumvent the multiple sheeted and singularity problems of implicit patches, BCX94a] introduces new sufficiency conditions for the BB form of trivariate polynomials within ....

....in Bernstein B ezier(BB) form defined within a tetrahedron[BCX94a] where the A stands for algebraic. Solutions to the problem of constructing a C 1 mesh of implicit algebraic patches based on polyhedron P have been given by [Dah89, BCX94a, BCX94b, DTS93, Guo91, Guo93, BI92] While papers [BI92, Dah89, DTS93, Guo91, Guo93] provide heuristics based on monotonicity and least square approximation to circumvent the multiple sheeted and singularity problems of implicit patches, BCX94a] introduces new sufficiency conditions for the BB form of trivariate polynomials within a tetrahedron, such that the zero contour of the ....

W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 181--193. Academic Press, Boston, 1989.

No context found.

W. Dahmen, Smooth piecewise quadratic surfaces, In Mathematical Methods in Computer Aided Geometric Design, T. Lyche, L.L. Schumaker editors, Academic Press, Boston, 1989, pp 181--193.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC