## Simple and Efficient Polygonization of Implicit Surfaces

Citations: | 29 - 2 self |

### BibTeX

@MISC{Velho_simpleand,

author = {Luiz Velho},

title = {Simple and Efficient Polygonization of Implicit Surfaces},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper describes a simple and efficient polygonization algorithm that gives a practical way to construct adapted piecewise linear representations of implicit surfaces. The method starts with a coarse uniform polygonal approximation of the surface and subdivides each polygon recursively according to local curvature. In that way, the inherent complexity of the problem is tamed by separating structuring from sampling and reducing part of the full three dimensional search to two dimensions.

### Citations

2073 | Marching cubes : A high resolution 3d surface construction algorithm
- Lorensen, Cline
- 1987
(Show Context)
Citation Context ...itial 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 m... |

569 | Decimation of triangle meshes
- Schroeder, Zarge, et al.
- 1992
(Show Context)
Citation Context ...of computing sub-optimal solutions. One attempt is this direction consists in first generating a fine uniform polygonization that is subsequently simplified by merging clusters of co-planar polygons, =-=[21]-=-. This two-step procedure reduces the algorithm complexity, but demands large memory resources and computing time. 5 A Simpler Adaptation Method In this paper we present an adaptive polygonization met... |

386 |
A Generalization of Algebraic Surface Drawing
- Blinn
- 1982
(Show Context)
Citation Context ...ns, such as quadrics, [6]. More complex implicit shapes can be specified by combining primitives using point set or blend operations that are the basis of, respectively, CSG, [18], and Blobby models, =-=[5]-=-. The implicit description is particularly effective to model smooth intricate shapes. For this reason, it has been extensively used in various areas of application. For example: in science for the vi... |

374 | Introduction to implicit surfaces
- Bloomenthal, Bajaj, et al.
- 1997
(Show Context)
Citation Context ...ivision structure balanced with repeated refinement steps that modifies only one cell at each pass. Whenever a cell is divided its neighbors are constrained to be at levels immediately above or below =-=[7]-=-, [14]. The second approach subdivide cells independently constraining polygon edges according to the adaptation criteria. In that way, it is able to make consistent decisions when it splits neighbor ... |

283 | Data structure for soft objects
- Wyvill, McPheeters, et al.
- 1986
(Show Context)
Citation Context ...est 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 co... |

223 |
Regular Polytopes
- Coxeter
- 1973
(Show Context)
Citation Context ...face and for each of these cells, it generates an element of the polygonal mesh approximating the surface. Our simplicial decomposition uses the classical Coxeter-Freudenthal space subdivision scheme =-=[10]-=-, that is defined as follows (see Figure 1). p0 p7 p1 p3 p2 p4 p5 p6 Figure 1: Coxeter-Freudenthal decomposition of the cube in R 3 The construction starts with a rectangular space partition of cubica... |

212 | A.G.: Representations For Rigid Solids: Theory, Methods And Systems
- Requicha
(Show Context)
Citation Context ...fied by algebraic functions, such as quadrics, [6]. More complex implicit shapes can be specified by combining primitives using point set or blend operations that are the basis of, respectively, CSG, =-=[18]-=-, and Blobby models, [5]. The implicit description is particularly effective to model smooth intricate shapes. For this reason, it has been extensively used in various areas of application. For exampl... |

79 | Collision detection by four-dimensional intersection testing
- Cameron
- 1990
(Show Context)
Citation Context ...ethods implementing various classes of operations with these objects. Typical examples are visualization operations, such as ray tracing, [20], and analysis operations, such as interference detection =-=[9]-=-. Although there are algorithms that perform such operations directly on the implicit form, sometimes, a conversion to another representation is necessary. An important case of conversion between geom... |

44 |
Adaptive polygonalization of implicitly defined surfaces
- Hall, Warren
- 1990
(Show Context)
Citation Context ...on structure balanced with repeated refinement steps that modifies only one cell at each pass. Whenever a cell is divided its neighbors are constrained to be at levels immediately above or below [7], =-=[14]-=-. The second approach subdivide cells independently constraining polygon edges according to the adaptation criteria. In that way, it is able to make consistent decisions when it splits neighbor cells ... |

35 | Physically-based methods for polygonization of implicit surfaces
- Figueiredo, Gomes, et al.
(Show Context)
Citation Context ...plicit function f . Under this paradigm, we interpret the modulus of f as a potential function and the gradient of jf j is used to generate a force field that drives particles to the implicit surface =-=[11]-=-. This results in the following equation of motion for a unit mass particle dx dt + sign(f)rf = 0 The above differential equation is solved iteratively using an explicit Euler time integration method.... |

34 | H.(1985).An algorithm for piecewise linear approximation of an implicitly de ned manifold
- Allgower, Schmidt
(Show Context)
Citation Context ...onization 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. A... |

17 |
Adaptive polygonization of implicit surfaces using simplicial decomposition and boundary constraints
- Velho
- 1990
(Show Context)
Citation Context ...roach subdivide cells independently constraining polygon edges according to the adaptation criteria. In that way, it is able to make consistent decisions when it splits neighbor cells sharing an edge =-=[22]-=-. All the algorithms described above employ a full 3D adaptive partition of space, and perform sampling a structuring in one single step. Strictly speaking, this three dimensional search is necessary ... |

9 | Computational morphology of curves
- Figueiredo, Gomes
- 1995
(Show Context)
Citation Context ...ions have polygon rendering engines that enable them to display these surfaces in real time. 2 Uniform Polygonization Polygonization methods incorporate two basic operations: sampling and structuring =-=[12]-=-. Sampling generates a set of points on the implicit surface. Structuring links those points to construct a mesh. The first operation deals with geometry, while the second operation deals with topolog... |

8 |
Practical uses for implicit surfaces in animation
- Beier
- 1990
(Show Context)
Citation Context ...application. For example: in science for the visualization of molecules [16]; in engineering for the design of industrial products, [19]; and in entertainment for the creation of animated characters, =-=[4]-=-. In order to do computations with implicitly defined objects a graphics system must process the implicit representation either in symbolic or 1 numeric form. The system must incorporate computational... |

4 |
A functional view of proteins
- Goodsell
- 1991
(Show Context)
Citation Context ...n is particularly effective to model smooth intricate shapes. For this reason, it has been extensively used in various areas of application. For example: in science for the visualization of molecules =-=[16]-=-; in engineering for the design of industrial products, [19]; and in entertainment for the creation of animated characters, [4]. In order to do computations with implicitly defined objects a graphics ... |

3 |
Using implicit surfaces to blend arbitrary solid models
- Rockwood, Owen
- 1987
(Show Context)
Citation Context .... For this reason, it has been extensively used in various areas of application. For example: in science for the visualization of molecules [16]; in engineering for the design of industrial products, =-=[19]-=-; and in entertainment for the creation of animated characters, [4]. In order to do computations with implicitly defined objects a graphics system must process the implicit representation either in sy... |

3 |
casting as a method for solid modeling
- Ray
- 1982
(Show Context)
Citation Context ...or 1 numeric form. The system must incorporate computational methods implementing various classes of operations with these objects. Typical examples are visualization operations, such as ray tracing, =-=[20]-=-, and analysis operations, such as interference detection [9]. Although there are algorithms that perform such operations directly on the implicit form, sometimes, a conversion to another representati... |

2 |
The algebraic properties of homogeneous second order surfaces
- Blinn
- 1984
(Show Context)
Citation Context ... Under this framework, a surface is defined as the set of points which satisfy the equation f(x; y; z) = 0. Simple primitive implicit shapes can be specified by algebraic functions, such as quadrics, =-=[6]-=-. More complex implicit shapes can be specified by combining primitives using point set or blend operations that are the basis of, respectively, CSG, [18], and Blobby models, [5]. The implicit descrip... |

1 |
Graphics Gems IV, volume 4, chapter An Implicit Surface Polygonizer
- Bloomenthal
- 1994
(Show Context)
Citation Context ...lem and it is a perfect match for our mesh adaptation method because of its simplicity and conciseness. The implementation of another uniform polygonization algorithm is described in Graphics Gems IV =-=[8]. Througho-=-ut this text, we use a "C-like" pseudo-code notation that is very close to the actual source code of the program. We define the data structure Vector, with the usual operators of dot product... |

1 | Adaptive polygonization of implicitly de�ned surfaces - Hall, Warren - 1990 |