E.L. Allgower and S. Gnutzmann. An algorithm for piecewise linear approximation of implicitly defined 2d surfaces. SIAM J. Numer. Anal., 24:452--469, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Beijing, 100080, China. Email: fxuguog cs.purdue.edu spatial partitioning to achieve a polygonal approximation. Hall and Warren [18] use a tetrahedral subdivision. Micchelli and Prautzsch [24] use uniform subdivision refinement algorithms for surface generation. Allgower and Gnutzmann, et al., [1], 2] use simplicial continuation or pivoting algorithms to generate a triangular or quadrilateral polygonal approximation. In this paper, we use neither space subdivision nor polyhedron continuation. Instead, we use a novel triangular expansion scheme on the surface starting from a seed point ....

....Let n(t) Dt E) 1 wt) be the normal function, where D, E 2 IR , w 2 IR. Then by (5.3) we have E = n 0 ; D = n 1 Gamma n 0 wn 1 Since the numerator of n (t) is a polynomial of degree 2 in t and n (t) 0 when t = 0 and t = 1, 5. 4) holds if there is another point in [0,1] such that (5.4) holds. Take t = then by )C ) 0, we have 1 w = Gamma (1) 0) Since C (0) B, C (1) 2A B = 2(p 1 Gamma p 0 ) Gamma B, it follows from (5.7) that 1 w = Gamma The good w should make 1 w 0, i.e. n(t) has no pole in [0,1] This ....

[Article contains additional citation context not shown here]

E.L. Allgower and S. Gnutzmann. An algorithm for piecewise linear approximation of implicitly defined two-dimensional surfaces. SIAM J. Numer. Anal., 24:452--469, 1987.

....triangles such that each blue point is covered by some triangle and no red point lies in any of the triangles. 1 Introduction In scientific computation, visualization, and computer graphics, the modeling and construction of surfaces is an important area. A small sample of some recent papers [2, 3, 5, 7, 10, 13, 20, 21] on this topic gives an indication of the scope and importance of this problem. The first author has been supported by National Science Foundation Grant CCR 93 01259 and an NYI award. Rather than delve into any specific problem studied in these papers, we focus on a general, abstract problem ....

E. Allgower and P. Schmidt, An algorithm for piecewise linear approximation of an implicitly defined manifold, SIAM J. Numerical Analysis 22 (1985), 322--346.

....triangles such that each blue point is covered by some triangle and no red point lies in any of the triangles. 1 Introduction In scientific computation, visualization, and computer graphics, the modeling and construction of surfaces is an important area. A small sample of some recent papers [2, 3, 5, 7, 10, 13, 20, 21] on this topic gives an indication of the scope and importance of this problem. The first author has been supported by National Science Foundation Grant CCR 93 01259 and an NYI award. Rather than delve into any specific problem studied in these papers, we focus on a general, abstract problem ....

E. Allgower and S. Gnutzmann, An algorithm for piecewise linear approximation of an implicitly defined two-dimensional surfaces, SIAM J. Numerical Analysis 24 (1987), 452--469.

....defined curves, has been successfully dealt with by conventional continuation methods (for example, 9] 16] 1] and [12] Codes exist for large classes of problems, for example AUTO [7] PITCON [16] PLTMG [3] etc. The literature on computing surfaces is not as well developed. Allgower [2] proposed an algorithm based on simplicial continuation, which describes the basic algorithm used in CAD and visualization (e.g. Marching Cubes [11] Rheinbolt [15] proposes using a smoothly varying projection of the tangent space onto the surface ( Moving Frame ) to wrap a grid onto the ....

E. L. Allgower and P. H. Schmidt, An algorithm for piecewise-linear approximation of an implicitly defined manifold, SIAM J. Numer. Anal., 22 (1985), pp. 322--346.

....approximations (polygons) based on space subdivision or polyhedron continuation. Bloomenthal [9] used octrees by spatial partition to reach a polygonal approximation. Micchelli and Prautzsch [16] used uniform refinement algorithms for surface generation. Allgower and Gnutzmann, et al., 3] 1] [2], used simplicial continuation or pivoting algorithms to generate a triangular or quadrilateral polygonal approximations. For the similar purpose, Chuang [11] used cubic continuation technique. In this paper, we use neither space subdivision nor polyhedron continuation. Instead, we use triangular ....

E.L. Allgower and S. Gnutzmann. An algorithm for piecewise linear approximation of implicitly defined two-dimensional surfaces. SIAM J. Numer. Anal., 24:452--469, 1987.

....the solution. 1 1.2 Previous Work The approximate conversion from implicit to parametric surfaces can be done through a polygonization method. This topic has been extensively researched in recent years with very good results (Wyvill, McPheeters and Wyvill, 1986) Lorensen and Cline, 1987) (Allgower and Gnutzmann, 1987), Bloomenthal, 1988) Velho, 1990) Hall and Warren, 1990) de Figueiredo et al. 1992) Polygonization algorithms construct a piecewise linear parametric description of the implicit surface that, among other things, is very useful for visualization purposes. Previous work on conversion from ....

Allgower, E. L. and Gnutzmann, S. (1987). An algorithm for piecewise linear approximation of implicitly defined two-dimensional surfaces. SIAM Journal of Numerical Analysis, 24(2):2452--2469.

....this section we describe the first step of the algorithm the computation of the initial mesh. This process corresponds to an uniform polygonization of the implicit surface which is a well understood problem and has various alternative solutions available in the graphics literature [15] 24] [2]. We describe here a polygonization algorithm based on a simplicial space decomposition [13] This is an elegant solution for the problem and it is a perfect match for our mesh adaptation method because of its simplicity and conciseness. The implementation of another uniform polygonization ....

....v[k] p.y = k 02) y : y inc; v[k] p. z = k 04) z : z inc; v[k] d = fvalue(v[k] p) v[k] n = fgrad(v[k] p) if (k = 0) side = sign(v[k] d) else if (side = sign(v[k] d) hit = TRUE; if (hit = TRUE) simplex(v[0] v[1] v[3] v[7] simplex(v[0] v[5] v[1] v[7] simplex(v[0] v[3] v[2],v[7] simplex(v[0] v[2] v[6] v[7] simplex(v[0] v[4] v[5] v[7] simplex(v[0] v[6] v[4] v[7] The procedure simplex determines, based on the sign of f , if there is an actual intersection with the cell, and if so, how it is pierced by the implicit surface. Note that, when the parent cube ....

[Article contains additional citation context not shown here]

E. L. Allgower and P. H. Schmidt. An algorithm for piecewise linear approximation of an implicitly defined manifold. SIAM Journal of Numerical Analysis, 22:322--346, 1985.

....can be classified according to how they implement these two operations. The simplest polygonization algorithms are based on uniform decompositions of the ambient space. They employ a cell complex made of, either cubical, or tetrahedral elements of the same size. See respectively [23] and [1]) The implicit function is evaluated at node points of the grid underlying the uniform spatial decomposition, and the samples obtained are assembled using adjacency relations from the cell complex. Algorithms of this type are straightforward to implement, but produce polygonal meshes that are not ....

....for (hit=FALSE, k=0; k 8; k ) v[k] p.x = k 01) x : x inc; v[k] p.y = k 02) y : y inc; v[k] p. z = k 04) z : z inc; v[k] d = fvalue(v[k] p) v[k] n = fgrad(v[k] p) if (k = 0) side = sign(v[k] d) else if (side = sign(v[k] d) hit = TRUE; if (hit = TRUE) simplex(v[0] v[1],v[3] v[7] simplex(v[0] v[5] v[1] v[7] simplex(v[0] v[3] v[2] v[7] simplex(v[0] v[2] v[6] v[7] simplex(v[0] v[4] v[5] v[7] simplex(v[0] v[6] v[4] v[7] The procedure simplex determines, based on the sign of f , if there is an actual intersection with the cell, and if so, how it is ....

[Article contains additional citation context not shown here]

E. L. Allgower and S. Gnutzmann. An algorithm for piecewise linear approximation of implicitly defined two-dimensional surfaces. SIAM Journal of Numerical Analysis, 24(2):2452--2469, April 1987.

....can be classified according to how they implement these two operations. The simplest polygonization algorithms are based on uniform decompositions of the ambient space. They employ a cell complex made of, either cubical, or tetrahedral elements of the same size. See respectively [18] and [1]) The implicit function is evaluated at node points of the grid underlying the uniform spatial decomposition, and the samples obtained are assembled using adjacency relations from the cell complex. Algorithms of this type are straightforward to implement, but produce polygonal meshes that are not ....

....for (hit=FALSE, k=0; k 8; k ) v[k] p.x = k 01) x : x inc; v[k] p.y = k 02) y : y inc; v[k] p. z = k 04) z : z inc; v[k] d = fvalue(v[k] p) v[k] n = fgrad(v[k] p) if (k = 0) side = sign(v[k] d) else if (side = sign(v[k] d) hit = TRUE; if (hit = TRUE) simplex(v[0] v[1],v[3] v[7] simplex(v[0] v[5] v[1] v[7] simplex(v[0] v[3] v[2] v[7] simplex(v[0] v[2] v[6] v[7] simplex(v[0] v[4] v[5] v[7] simplex(v[0] v[6] v[4] v[7] The procedure simplex determines, based on the sign of f , if there is an actual intersection with the cell, and if so, how it is ....

[Article contains additional citation context not shown here]

E. L. Allgower and S. Gnutzmann. An algorithm for piecewise linear approximation of implicitly defined two-dimensional surfaces. SIAM Journal of Numerical Analysis, 24(2):2452--2469, April 1987.

....from the normal of the polygon s support plane. Below, we explain each step in detail. 6. 1 Initial Mesh This process corresponds to an uniform polygonization of the implicit surface, which is a well understood problem and has various alternative solutions available in the graphics literature [32 35]. We employ a polygonization algorithm based on a simplicial space decomposition [36] This is an elegant solution for the problem and it is a perfect match for our mesh adaptation method because of its simplicity and conciseness. The uniform polygonization algorithm decomposes the bounding box ....

E. L. Allgower and P. H. Schmidt. An algorithm for piecewise linear approximation of an implicitly defined manifold. SIAM Journal of Numerical Analysis, 22:322--346, 1985.

....manifold cannot be triangulated if it is not differentiable. In the last two decades a great deal of attention has been given to simplicial continuation methods. Much of this effort has come from or been integrated by Allgower and colleagues (Allgower and Georg 1980; Allgower and Schmidt 1985; Allgower and Gnutzmann 1987; Allgower and Georg 1990; Allgower and Gnutzmann 1991) In particular, Allgower and Gnutzmann applied simplicial continuation methods to generate piecewise linear approximations to implicit surfaces (Allgower and Gnutzmann 1987; Allgower and Gnutzmann 1991) These two papers describe algorithms ....

.... and colleagues (Allgower and Georg 1980; Allgower and Schmidt 1985; Allgower and Gnutzmann 1987; Allgower and Georg 1990; Allgower and Gnutzmann 1991) In particular, Allgower and Gnutzmann applied simplicial continuation methods to generate piecewise linear approximations to implicit surfaces (Allgower and Gnutzmann 1987; Allgower and Gnutzmann 1991) These two papers describe algorithms for continuation methods and simplicial tilings of space. Their 1991 paper makes important contributions to the display of surface triangulations: they show how to make a simplicial mesh uniform; they show how to refine a mesh ....

Allgower. E. L. and S. Gnutzmann. 1987. An algorithm for piecewise-linear approximation of implicitly defined two-dimensional surfaces. SIAM Journal on Numerical Analysis, 24(1): 452-469.

....cross a critical boundary. The standard procedure to study the structure of the solution set of (1) is to follow one dimensional paths on the solution surfaces. There are only a few papers that discuss the computation of grids of solution points. Allgower and Schmidt [1] Allgower and Gnutzmann [2] developed simplicial continuation algorithms to triangulate the solution surface. Their algorithms are only suited for small problems. Rheinboldt [12] developed the moving frame algorithm that can be used for moderately large cases of (1) This algorithm computes a smooth orthogonal basis of ....

E. L. Allgower and S. Gnutzmann, An algorithm for piecewise linear approximation of implicitly dened two-dimensional surfaces, SIAM J. Numer. Anal., 24 (1987), pp. 452-469.

....lose or gain stability as we cross a critical boundary. The standard procedure to study the structure of the solution set of (1) is to follow one dimensional paths on the solution surfaces. There are only a few papers that discuss the computation of grids of solution points. Allgower and Schmidt [1], Allgower and Gnutzmann [2] developed simplicial continuation algorithms to triangulate the solution surface. Their algorithms are only suited for small problems. Rheinboldt [12] developed the moving frame algorithm that can be used for moderately large cases of (1) This algorithm computes a ....

E. L. Allgower and P. H. Schmidt, An algorithm for piecewise-linear approximation of an implicitly dened manifold, SIAM J. Numer. Anal., 22 (1985), pp. 322-346.

....of # do begin obtain # from # by pivoting across # ; if # # # then #: ## # ; check whether # is new end for ; print #; # : # # ; output of checked intersecting tetrahedra end while . The above algorithm is an adaptation of an algorithm for approximating the boundary of D, see, e.g. [1, 3, 4] and most recently in [16] In general it is trivial to obtain a starting tetrahedron: one only needs to know one point in D. The properties of standard triangulations such as the Coxeter Freudenthal triangulation permit a compact storing, retrieval and comparing of the tetrahedra in the list #. ....

E. L. Allgower and P. H. Schmidt, An algorithm for piecewise-linear approximation of an implicitly defined manifold, SIAM J. Numer. Anal., 22 (1985), pp. 322--346.

No context found.

Allgower, E. L. & Gnutzmann, S. (1987): An algorithm for piecewise linear approximation of implicitly defined two-dimensional surfaces. SIAM J. Numer. Anal. 24, 452--469.

....twice di erentiable map. Let N : n m. Assume that the ber F : H 1 (y 0 ) of H over a point y 0 2 R m is a compact di erentiable manifold, and that H is of maximal rank in a neighborhood of F . Thus the dimension of F is n. Without loss of generality we can assume y 0 is the origin 0. In [4, 5, 6] a numerical method to compute a piecewise linear structure on F is developed. Since it depends on a triangulation of a neighborhood of F in the ambient R N , the piecewise linear approximation is global, however the combinatorics associated with the simplices of the triangulation is of order ....

....Lemma 1.1 that if m n 1, the projection : R n m R 2n 1 onto the rst 2n 1 coordinates gives an embedding of F . Let e F : F ) We assume that we are in the case N : n m 2n 1, since otherwise the result is done already by Allgower and Gnutzmann [4, 5] and Allgower and Schmidt [6]. Programs in C for low values of n are available via the URL site www.math.colostate.edu georg . The following lemma gives a key fact about e F . Lemma 2.1. Let H, N : n m, F , and e F be as at the start of this section. There exists a C 2 map e H : R 2n 1 R n 1 with e H 1 (0) ....

[Article contains additional citation context not shown here]

E. L. Allgower and Ph. H. Schmidt. An algorithm for piecewise-linear approximation of an implicitly dened manifold. SIAM J. Numer. Anal., 22:322-346, 1985.

....twice di erentiable map. Let N : n m. Assume that the ber F : H 1 (y 0 ) of H over a point y 0 2 R m is a compact di erentiable manifold, and that H is of maximal rank in a neighborhood of F . Thus the dimension of F is n. Without loss of generality we can assume y 0 is the origin 0. In [4, 5, 6] a numerical method to compute a piecewise linear structure on F is developed. Since it depends on a triangulation of a neighborhood of F in the ambient R N , the piecewise linear approximation is global, however the combinatorics associated with the simplices of the triangulation is of order ....

....generically, it follows from Lemma 1.1 that if m n 1, the projection : R n m R 2n 1 onto the rst 2n 1 coordinates gives an embedding of F . Let e F : F ) We assume that we are in the case N : n m 2n 1, since otherwise the result is done already by Allgower and Gnutzmann [4, 5] and Allgower and Schmidt [6] Programs in C for low values of n are available via the URL site www.math.colostate.edu georg . The following lemma gives a key fact about e F . Lemma 2.1. Let H, N : n m, F , and e F be as at the start of this section. There exists a C 2 map e H : R ....

E. L. Allgower and S. Gnutzmann. An algorithm for piecewise linear approximation of implicitly dened two-dimensional surfaces. SIAM J. Numer. Anal., 24:452-469, 1987.

....defined as a solution set of an equation H(x) 0, where 0 # range H and H :IR N K # IR N is a smooth map. In this paper we derive error estimates and global approximation results for the algorithms given in [2] and [5] Analogous results can also be obtained for the algorithms given in [3] and [7] Many of the given error estimates have been obtained by Saigal[8] for the case K = 0. However for K 0, the proofs need a di#erent approach. Although our assumptions can be relaxed in several ways, we make them here in order to hold our discussion to a simple and unified form. ....

E. L. Allgower & Ph. H. Schmidt, An algorithm for piecewise-linear approximation of an implicitly defined manifold, SIAM J. Numer. Anal. 22 (1985), 322--346.

....obtaining piecewise linear (PL) approximations of manifolds M which are implicitly defined as a solution set of an equation H(x) 0, where 0 # range H and H :IR N K # IR N is a smooth map. In this paper we derive error estimates and global approximation results for the algorithms given in [2] and [5] Analogous results can also be obtained for the algorithms given in [3] and [7] Many of the given error estimates have been obtained by Saigal[8] for the case K = 0. However for K 0, the proofs need a di#erent approach. Although our assumptions can be relaxed in several ways, we make ....

....to H. By this we mean that H 1 T (##) # # #= # for small # 0 where ## : #, # 2 , # N ) t . In the special case K = 0, such simplices are termed completely labeled , see e.g. 1] It can be seen [4] that for almost all # 0, ## is a regular value of H T , and hence, see [2] and [5] for almost all # 0, the family M T : H 1 T (##) # # # # # # T transverse is a K dimensional PL manifold consisting of polytopes. Our aim is to estimate the quality with which M T approximates M . In the sequel we use the following notation and assumptions: 1.1) The ....

[Article contains additional citation context not shown here]

E. L. Allgower & S. Gnutzmann, An algorithm for piecewise linear approximation of implicitly defined two-dimensional surfaces, SIAM J. Numer. Anal. 24 (1987), 452--469.

No context found.

E.L. Allgower and S. Gnutzmann. An algorithm for piecewise linear approximation of implicitly defined 2d surfaces. SIAM J. Numer. Anal., 24:452--469, 1987.

No context found.

Eugene L. Allgower and Phillip H. Schmidt. An algorithm for piecewise-linear approximation of an implicitly defined manifold. SIAM Journal on Numerical Analysis, 22(2):322--346, April 1985.

No context found.

E. L. Allgower and P. H. Schmidt. An algorithm for piecewise linear approximation of an implicitly defined manifold. SIAM Journal of Numerical Analysis, 22:322--346, April 1985.

No context found.

E. L. Allgower and P. H. Schmidt. An algorithm for piecewise linear approximation of an implicitly defined manifold. SIAM Journal of Numerical Analysis, 22:322--346, April 1985.

No context found.

E. L. Allgower and P. H. Schmidt, An algorithm for piecewise-linear approximation of an implicitly defined manifold, SIAM Journal on Numerical Analysis

No context found.

E. L. Allgower and S. Gnutzmann. An algorithm for piecewise linear approximation of implicitly defined two-dimensional surfaces. SIAM Journal of Numerical Analysis, 24(2):2452--2469, April 1987.

First 50 documents  Next 50

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