## Parametrization and smooth approximation of surface triangulations (1997)

### Cached

### Download Links

- [folk.uio.no]
- [www.cs.jhu.edu]
- [heim.ifi.uio.no]
- [www.ifi.uio.no]
- [www.seas.upenn.edu]
- [www.oslo.sintef.no]
- [www.ann.jussieu.fr]
- [www.multires.caltech.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | Computer Aided Geometric Design |

Citations: | 254 - 15 self |

### BibTeX

@ARTICLE{Floater97parametrizationand,

author = {Michael S. Floater},

title = {Parametrization and smooth approximation of surface triangulations},

journal = {Computer Aided Geometric Design},

year = {1997},

volume = {14},

pages = {231--250}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. A method based on graph theory is investigated for creating global parametrizations for surface triangulations for the purpose of smooth surface fitting. The parametrizations, which are planar triangulations, are the solutions of linear systems based on convex combinations. A particular parametrization, called shape-preserving, is found to lead to visually smooth surface approximations. A standard approach to fitting a smooth parametric curve c(t) through a given sequence of points xi = (xi,yi,zi) ∈ IR 3, i = 1,...,N is to first make a parametrization, a corresponding increasing sequence of parameter values ti. By finding smooth functions x,y,z: [t1,tN] → IR for which x(ti) = xi, y(ti) = yi, z(ti) = zi, an interpolatory curve

### Citations

1098 |
A Practical Guide to Splines
- Boor
- 1978
(Show Context)
Citation Context ...)) results. Two commonly used parametrizations are the uniform and chord length ones, in which and (ti+1 − ti)/(ti − ti−1) = 1, (1) (ti+1 − ti)/(ti − ti−1) = �xi+1 − xi�/�xi − xi−1�, (2) respectively =-=[2]-=-, [6]. In this paper we investigate, by way of numerical examples, a method for making parametrizations for a surface triangulation S (having triangular facets and a boundary), based on a method propo... |

903 |
Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide
- Farin
- 1990
(Show Context)
Citation Context ...sults. Two commonly used parametrizations are the uniform and chord length ones, in which and (ti+1 − ti)/(ti − ti−1) = 1, (1) (ti+1 − ti)/(ti − ti−1) = �xi+1 − xi�/�xi − xi−1�, (2) respectively [2], =-=[6]-=-. In this paper we investigate, by way of numerical examples, a method for making parametrizations for a surface triangulation S (having triangular facets and a boundary), based on a method proposed b... |

651 | Surface reconstruction from unorganized points
- Hoppe, DeRose, et al.
- 1992
(Show Context)
Citation Context ...ce triangulations also arise as crude approximations of various kinds of nonexplicit data sets such as cross-sectional data [22], [9] and densely scattered data sampled from three-dimensional objects =-=[13]-=-, provided the topology is suitable. In these cases usually no parametrization is available and it is especially important to be able to construct one. §4. Method for parametrization Finding any param... |

454 | Computational Geometry
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...mple 1. Let u1,...,uN, N ≥ 3, be arbitrary pairwise distinct points in IR 2 , not all colinear, and let z1,...,zN ∈ IR be arbitrary. Let P be any triangulation, for example the Delaunay triangulation =-=[20]-=-, [21], of the ui and set xi = (ui,vi,zi), for i = 1,...,N, where ui = (ui,vi). Then P is a planar triangulation and S(P,X) is a surface triangulation with parametrization P. In Example 1, a natural p... |

383 | The factors of graphs
- Tutte
- 1952
(Show Context)
Citation Context ...s paper we investigate, by way of numerical examples, a method for making parametrizations for a surface triangulation S (having triangular facets and a boundary), based on a method proposed by Tutte =-=[25]-=- for making straight line drawings of planar graphs. The nodes xi ∈ IR 3 of S are mapped to points ui = (ui,vi) ∈ D, for some convex D ⊂ IR 2 , in such a way that the image of S is a planar triangulat... |

313 |
An Introduction to Numerical Analysis
- Atkinson
- 1989
(Show Context)
Citation Context ...(u,v) ↦→ (u,v,f(u,v)) by a parametric surface s : IR 2 → IR 3 . This allows the possibility of choosing other parametrizations of S which may yield smoother surface approximations. Example 2. Let s : =-=[0,1]-=-×[0,1] → IR 3 be a smooth surface and let u1,...,uN, N ≥ 3, be arbitrary pairwise distinct points in [0,1]×[0,1], not all colinear. Let P be any triangulation of the ui. If xi = s(ui), X = {xi : i = 1... |

286 |
Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems
- Vorst
- 1992
(Show Context)
Citation Context ...the non-zero entries in a diagonal band structure. This kind of matrix occurs frequently in the numerical solution of differential equations and it was found that an iterative method called Bi-CGSTAB =-=[26]-=- was highly effective. This is a variant of the conjugate gradient method for non-symmetric matrices. Bi-CGSTAB was used to solve each equation in (10) for n of the order of 25000, setting every inter... |

152 | Free-form shape design using triangulated surfaces
- Welch, Witkin
- 1994
(Show Context)
Citation Context ...the triangular facets in Si. However both these methods can be unstable when there are large angles between the triangular facets in Si. We have adopted instead a neat method, due to Welch and Witkin =-=[27]-=-, for making local parametrizations for surface triangulations which emulates the so-called geodesic polar map, a local mapping known in differential geometry which preserves arc length in each radial... |

124 | On straight lines representation of planar graphs - Fáry - 1948 |

89 |
Data dependent triangulations for piecewise linear interpolation
- Dyn, Levin, et al.
- 1990
(Show Context)
Citation Context ...ce triangulations. Such optimizations are usually based on recursively swapping edges in an existing one according to some goodness criterion as described by Schumaker [21], and Dyn, Levin, and Rippa =-=[5]-=-. Moreover replacing P by ˆ P may have a beneficial effect on the surface approximation when it is based on piecewise polynomials on ˆ P. In Figure 17 the Delaunay triangulation ˆP, which maximizes th... |

79 |
Convex representations of graphs
- Tutte
- 1960
(Show Context)
Citation Context ...s a matter of constructing a straight line drawing of a plane graph. It was first shown by Fáry [8], using a proof by induction, that every (simple) plane graph has a straight line drawing. Later, in =-=[24,25]-=-, Tutte gave a constructive proof employing a method known as the ‘barycentric mapping’ for making such a drawing. The boundary nodes of G are mapped to the boundary of any convex polygon and every in... |

76 |
Triangular Bernstein-Bezier patches
- Farin
- 1986
(Show Context)
Citation Context ...y in the first and by chord length in the second two. In each case three C 1 piecewise-cubic interpolants x, y, z, satisfying (3), were then made for each component using the Clough Tocher split on P =-=[7]-=-, [11]. The resulting interpolatory surface s(u,v), of the form (4) was then sampled on a square 40×40 grid and interpolated by a C 2 cubic tensor-product spline s ′ (u,v). These are shown in Figures ... |

32 |
Triangular Bernstein-Bézier patches, Computer Aided Geometric Design
- Farin
- 1986
(Show Context)
Citation Context ...re mapped uniformly in the first and by chord length in the second two. In each case three C 1 piecewise-cubic interpolants x, y, z, satisfying (3), were then made using the Clough Tocher split on P (=-=Farin, 1986-=-; Franke and Schumaker, 1986). The resulting interpolatory surface s(u, v), of the form (4) was then sampled on a square 40 × 40 grid and interpolated by a C 2 cubic tensor product spline s ′ (u, v). ... |

21 |
Knot selection for parametric spline interpolation
- Foley, Nielson
- 1992
(Show Context)
Citation Context ... known that choosing parameter values by chord length tends to lead to smoother curve approximations than using uniform especially when the data points are unevenly distributed; see Foley and Nielson =-=[10]-=-. For this reason we search for something analogous for surface triangulations. First observe that there are other ways of determining the ti from the Li. Proposition 5. Suppose t1 < t2 < · · · tN, ti... |

18 |
Reconstructing 3D objects from cross-sections
- Schumaker
- 1990
(Show Context)
Citation Context ... 2 need not be injectively projectable onto the plane as in Example 1. Surface triangulations also arise as crude approximations of various kinds of nonexplicit data sets such as cross-sectional data =-=[22]-=-, [9] and densely scattered data sampled from three-dimensional objects [13], provided the topology is suitable. In these cases usually no parametrization is available and it is especially important t... |

13 |
Triangulation methods, in Topics in Multivariate Approximation
- Schumaker
- 1987
(Show Context)
Citation Context .... Let u1,...,uN, N ≥ 3, be arbitrary pairwise distinct points in IR 2 , not all colinear, and let z1,...,zN ∈ IR be arbitrary. Let P be any triangulation, for example the Delaunay triangulation [20], =-=[21]-=-, of the ui and set xi = (ui,vi,zi), for i = 1,...,N, where ui = (ui,vi). Then P is a planar triangulation and S(P,X) is a surface triangulation with parametrization P. In Example 1, a natural paramet... |

9 |
Applied Graph Theory
- Marshall
- 1971
(Show Context)
Citation Context ...t in [14] is a numerical approximation, using finite elements or finite differences of elliptic equations. §2. Graphs and triangulations We shall draw some standard definitions from graph theory; see =-=[17]-=-. A (simple) graph G = G(V,E) is a set of nodes V = {i : i = 1,...,N} and a set of edges E, a subset of the set of all unordered pairs of nodes (i,j), (i �= j). A node j is a neighbour of node i if (i... |

8 |
A storage-efficient method for construction of a Thiessen triangulation
- Cline, Renka
- 1984
(Show Context)
Citation Context ...l guess. The algorithm could be stopped after only a few hundred iterations and no instabilities were experienced. A suitable data structure for the triangulations is that proposed by Cline and Renka =-=[4]-=- for efficient storage of triangulations. Figure 6 shows a Delaunay triangulation of a set of 27 scattered data points in the plane. The points were mapped uniformly (λi,j = 1/di for (i,j) ∈ E) into t... |

8 |
Parametric surface interpolation
- Lounsberry, Mann, et al.
- 1992
(Show Context)
Citation Context ...ngulations fit one or more piecewise polynomial patches to each triangular facet so that the overall geometric continuity is G k , for some k. Such methods are surveyed by Lounsberry, Mann and DeRose =-=[15]-=-. However this will generate a large amount of data when the number of triangles is high. If the triangulation is a single patch it may be preferable to approximate it with a single parametric surface... |

7 |
A bibliography of multivariate approximation
- Franke, Schumaker
- 1986
(Show Context)
Citation Context ...) and the second and third both generalize (2). The name ‘shape-preserving’ is chosen because this parametrization has a reproduction property (Proposition 6). Using a suitable scattered data method (=-=Franke and Schumaker, 1986-=-), smooth functions x, y, z : D → R, can be constructed independently to satisfy x(ui)=xi, y(ui)=yi, z(ui)=zi. (3) A parametric surface s: D → R 3 satisfying s(ui)=xi then results from setting s(u)= �... |

7 | G 1 continuity of B-spline surface patches in reverse engineering - Milroy, Bradley, et al. - 1995 |

6 |
Triangulation methods
- Schumaker
- 1987
(Show Context)
Citation Context ..., be arbitrary pairwise distinct points in R 2 , not all collinear, and let z1,...,zN ∈ R be arbitrary. Let P be any triangulation, for example the Delaunay triangulation (Preparata and Shamos, 1985; =-=Schumaker, 1987-=-), of the ui and set xi =(ui,vi,zi),fori=1,...,N,whereui =(ui,vi).ThenPis a planar triangulation and S(P, X) is a surface triangulation with parametrization P. In Example 1, a natural parametrization ... |

5 | Smooth surface reconstruction from cross-sections using implicit methods
- Floater, Westgaard
- 1992
(Show Context)
Citation Context ...d not be injectively projectable onto the plane as in Example 1. Surface triangulations also arise as crude approximations of various kinds of nonexplicit data sets such as cross-sectional data [22], =-=[9]-=- and densely scattered data sampled from three-dimensional objects [13], provided the topology is suitable. In these cases usually no parametrization is available and it is especially important to be ... |

5 |
Harmonic models of shape transformations in digital images and patterns, CVGIP: graph. models and imag
- Li, Suen, et al.
- 1992
(Show Context)
Citation Context ...Milroy et al. [18] have made such approximations directly via a non-linear minimization. Techniques for mapping and transforming triangulations have been proposed by Maillot et al. [16] and Li et al. =-=[14]-=-. The method in [16] is to minimize an energy functional based on the theory of elasticity, while that in [14] is a numerical approximation, using finite elements or finite differences of elliptic equ... |

4 |
Drawing plane graphs nicely
- Chiba, Onoguchi, et al.
- 1985
(Show Context)
Citation Context ...the boundary of any convex polygon and every internal node in the drawing is defined to be the centre of mass of its neighbours, their barycentre; see Figure 2. The reader is referred to Chiba et al. =-=[3]-=- and Nishizeki [19] for discussions and further methods for constructing straight line drawings of planar graphs. Tutte’s barycentric mapping is the start point for the remainder of this paper. In the... |

3 |
A bibliography of multivariate approximation, in Topics in Multivariate Approximation
- Franke, Schumaker
- 1986
(Show Context)
Citation Context ...1) and the second and third both generalize (2). The name ‘shape-preserving’ is chosen because this parametrization has a reproduction property (Proposition 6). Using a suitable scattered data method =-=[11]-=-, smooth functions x,y,z : D → IR, can be constructed independently to satisfy x(ui) = xi, y(ui) = yi, z(ui) = zi. (3) A parametric surface s : D → IR 3 satisfying s(ui) = xi then results from setting... |

3 |
Harmonic models of shape transformations
- Li, Suen, et al.
- 1992
(Show Context)
Citation Context ...ngulations have been proposed by Maillot et al. (1993) and Li et al. (1992). The method in (Maillot et al., 1993) is to minimize an energy functional based on the theory of elasticity, while that in (=-=Li et al., 1992-=-) is a numerical approximation, using finite elements or finite differences of elliptic equations. 2. Graphs and triangulations We shall draw some standard definitions from graph theory; see (Marshall... |

1 |
On straight line representatation of planar graphs
- Fary
- 1948
(Show Context)
Citation Context ...ght line edges) isomorphic to a given triangulated plane graph G. In the language of graph theory this is a matter of constructing a straight line drawing of a plane graph. It was first shown by Fáry =-=[8]-=-, using a proof by induction, that every (simple) plane graph has a straight line drawing. Later, in [24,25], Tutte gave a constructive proof employing a method known as the ‘barycentric mapping’ for ... |

1 |
G 1 continuity of B-spline surface patches
- Milroy, Bradley, et al.
- 1995
(Show Context)
Citation Context ...hen the number of triangles is high. If the triangulation is a single patch it may be preferable to approximate it with a single parametric surface, for example a tensor-product spline. Milroy et al. =-=[18]-=- have made such approximations directly via a non-linear minimization. Techniques for mapping and transforming triangulations have been proposed by Maillot et al. [16] and Li et al. [14]. The method i... |

1 |
Planar graph problems, Computing Supplementum 7
- Nishizeki
- 1990
(Show Context)
Citation Context ..., (b) each edge (i,j) ∈ E is mapped to a curve whose endpoints are i and j, (c) the only intersections between curves are at common endpoints. Such a planar embedding of G is referred to by Nishizeki =-=[19]-=- as a plane graph. A plane graph partitions the plane into connected regions called faces. In particular the unbounded face is called the outer face. Different embeddings of a planar graph may partiti... |

1 | Floater / Computer Aided Geometric Design 14 - S - 1997 |

1 | A storage-e cient method for construction of a Thiessen triangulation - Cline, Renka - 1984 |

1 | A bibliography ofmultivariate approximation, in Topics in Multivariate Approximation - Franke, Schumaker - 1986 |

1 | Triangular Bernstein-Brzier patches, Computer Aided Geometric Design - Farin - 1986 |

1 | On straight line representation of planar graphs - Ffiry - 1948 |