The Boundary Representation of a 3D manifold contains
FACES (connected subsets of a parametric surface S : R2 -!
R3). In many science and engineering applications it is cumbersome
and algebraically difficult to deal with the polynomial set and
constraints (LOOPs) representing the FACE. Because of this reason, a
Piecewise Linear (PL) approximation of the FACE is needed, which is
usually represented in terms of triangles (i.e. 2-simplices). Solving the
problem of FACE triangulation requires producing quality triangles
which are: (i) independent of the arguments of S, (ii) sensitive to the
local curvatures, and (iii) compliant with the boundaries of the FACE
and (iv) topologically compatible with the triangles of the neighboring
FACEs. In the existing literature there are no guarantees for the point
(iii). This article contributes to the topic of triangulations conforming
to the boundaries of the FACE by applying the concept of parameterindependent
Gabriel complex, which improves the correctness of the
triangulation regarding aspects (iii) and (iv). In addition, the article
applies the geometric concept of tangent ball to a surface at a point to
address points (i) and (ii). Additional research is needed in algorithms
that (i) take advantage of the concepts presented in the heuristic
algorithm proposed and (ii) can be proved correct.
[1] K. Abe, J. Bisceglio, T. J. Peters, A. C. Russell, and T. Sakkalis.
Computational topology for reconstruction of surfaces with boundary:
Integrating experiments and theory. In SMI -05: Proceedings of the
International Conference on Shape Modeling and Applications 2005,
pages 290-299, Washington, DC, USA, 2005. IEEE Computer Society.
[2] Udo Adamy, Joachim Giesen, and Matthias John. New techniques for
topologically correct surface reconstruction. In VIS -00: Proceedings of
the conference on Visualization -00, pages 373-380, Los Alamitos, CA,
USA, 2000. IEEE Computer Society Press.
[3] N. Amenta, M. Bern, and D. Eppstein. The crust and the betaskeleton:
Combinatorial curve reconstruction. Graphical models and
image processing: GMIP, 60(2):125-, 1998.
[4] Nina Amenta and Marshall Bern. Surface reconstruction by voronoi
filtering. In SCG -98: Proceedings of the fourteenth annual symposium
on Computational geometry, pages 39-48, New York, NY, USA, 1998.
ACM.
[5] Dominique Attali, Jean-Daniel Boissonnat, and Andr'e Lieutier. Complexity
of the delaunay triangulation of points on surfaces the smooth
case. In SCG -03: Proceedings of the nineteenth annual symposium on
Computational geometry, pages 201-210, New York, NY, USA, 2003.
ACM.
[6] J-D Boissonnat and S. Oudot. An effective condition for sampling
surfaces with guarantees. In SM -04: Proceedings of the ninth ACM
symposium on Solid modeling and applications, pages 101-112, Airela-
Ville, Switzerland, Switzerland, 2004. Eurographics Association.
[7] Jean-Daniel Boissonnat and Steve Oudot. Provably good sampling and
meshing of lipschitz surfaces. In SCG -06: Proceedings of the twentysecond
annual symposium on Computational geometry, pages 337-346,
New York, NY, USA, 2006. ACM.
[8] Manfredo Do Carmo. Differential geometry of curves and surfaces,
pages 1-168. Prentice Hall, 1976. ISBN: 0-13-212589-7.
[9] Siu-Wing Cheng, Tamal K. Dey, Edgar A. Ramos, and Tathagata
Ray. Sampling and meshing a surface with guaranteed topology and
geometry. In SCG -04: Proceedings of the twentieth annual symposium
on Computational geometry, pages 280-289, New York, NY, USA, 2004.
ACM.
[10] L. Paul Chew. Guaranteed-quality mesh generation for curved surfaces.
In SCG -93: Proceedings of the ninth annual symposium on Computational
geometry, pages 274-280, New York, NY, USA, 1993. ACM.
[11] Museo del Oro Bogot'a DC. Pre-columbian fish.
http://www.banrep.org/museo/eng/home4.htm.
[12] Herbert Edelsbrunner and Nimish R. Shah. Triangulating topological
spaces. In SCG -94: Proceedings of the tenth annual symposium on
Computational geometry, pages 285-292, New York, NY, USA, 1994.
ACM.
[13] Rosalinda Ferrandes. Pump carter.
http://shapes.aimatshape.net/viewgroup.php?id=919.
[14] Rosalinda Ferrandes. Stub axle.
http://shapes.aimatshape.net/viewgroup.php?id=917.
[15] Greg Leibon and David Letscher. Delaunay triangulations and voronoi
diagrams for riemannian manifolds. In SCG -00: Proceedings of the
sixteenth annual symposium on Computational geometry, pages 341-
349, New York, NY, USA, 2000. ACM.
[16] A' ngel Montesdeoca. Apuntes de geometr'─▒a diferencial de curvas y
superficies. Santa Cruz de Tenerife, 1996. ISBN: 84-8309-026-0.
[17] Oscar E. Ruiz, John Congote, Carlos Cadavid, Juan G. Lalinde, and
Guillermo Peris. Parameter-independent, curvature-sensitive sprinkle
and star algorithms for surface triangulation. Submitted to the Computer-
Aided Design Journal at Elsevier editorial.
[18] Oscar E. Ruiz and Sebastian Pe˜na. Aspect ratio- and size-controlled
patterned triangulations of parametric surfaces. In Ninth IASTED Intl.
Conf. Computer Graphics and Imaging, February 13-15 2007.
[1] K. Abe, J. Bisceglio, T. J. Peters, A. C. Russell, and T. Sakkalis.
Computational topology for reconstruction of surfaces with boundary:
Integrating experiments and theory. In SMI -05: Proceedings of the
International Conference on Shape Modeling and Applications 2005,
pages 290-299, Washington, DC, USA, 2005. IEEE Computer Society.
[2] Udo Adamy, Joachim Giesen, and Matthias John. New techniques for
topologically correct surface reconstruction. In VIS -00: Proceedings of
the conference on Visualization -00, pages 373-380, Los Alamitos, CA,
USA, 2000. IEEE Computer Society Press.
[3] N. Amenta, M. Bern, and D. Eppstein. The crust and the betaskeleton:
Combinatorial curve reconstruction. Graphical models and
image processing: GMIP, 60(2):125-, 1998.
[4] Nina Amenta and Marshall Bern. Surface reconstruction by voronoi
filtering. In SCG -98: Proceedings of the fourteenth annual symposium
on Computational geometry, pages 39-48, New York, NY, USA, 1998.
ACM.
[5] Dominique Attali, Jean-Daniel Boissonnat, and Andr'e Lieutier. Complexity
of the delaunay triangulation of points on surfaces the smooth
case. In SCG -03: Proceedings of the nineteenth annual symposium on
Computational geometry, pages 201-210, New York, NY, USA, 2003.
ACM.
[6] J-D Boissonnat and S. Oudot. An effective condition for sampling
surfaces with guarantees. In SM -04: Proceedings of the ninth ACM
symposium on Solid modeling and applications, pages 101-112, Airela-
Ville, Switzerland, Switzerland, 2004. Eurographics Association.
[7] Jean-Daniel Boissonnat and Steve Oudot. Provably good sampling and
meshing of lipschitz surfaces. In SCG -06: Proceedings of the twentysecond
annual symposium on Computational geometry, pages 337-346,
New York, NY, USA, 2006. ACM.
[8] Manfredo Do Carmo. Differential geometry of curves and surfaces,
pages 1-168. Prentice Hall, 1976. ISBN: 0-13-212589-7.
[9] Siu-Wing Cheng, Tamal K. Dey, Edgar A. Ramos, and Tathagata
Ray. Sampling and meshing a surface with guaranteed topology and
geometry. In SCG -04: Proceedings of the twentieth annual symposium
on Computational geometry, pages 280-289, New York, NY, USA, 2004.
ACM.
[10] L. Paul Chew. Guaranteed-quality mesh generation for curved surfaces.
In SCG -93: Proceedings of the ninth annual symposium on Computational
geometry, pages 274-280, New York, NY, USA, 1993. ACM.
[11] Museo del Oro Bogot'a DC. Pre-columbian fish.
http://www.banrep.org/museo/eng/home4.htm.
[12] Herbert Edelsbrunner and Nimish R. Shah. Triangulating topological
spaces. In SCG -94: Proceedings of the tenth annual symposium on
Computational geometry, pages 285-292, New York, NY, USA, 1994.
ACM.
[13] Rosalinda Ferrandes. Pump carter.
http://shapes.aimatshape.net/viewgroup.php?id=919.
[14] Rosalinda Ferrandes. Stub axle.
http://shapes.aimatshape.net/viewgroup.php?id=917.
[15] Greg Leibon and David Letscher. Delaunay triangulations and voronoi
diagrams for riemannian manifolds. In SCG -00: Proceedings of the
sixteenth annual symposium on Computational geometry, pages 341-
349, New York, NY, USA, 2000. ACM.
[16] A' ngel Montesdeoca. Apuntes de geometr'─▒a diferencial de curvas y
superficies. Santa Cruz de Tenerife, 1996. ISBN: 84-8309-026-0.
[17] Oscar E. Ruiz, John Congote, Carlos Cadavid, Juan G. Lalinde, and
Guillermo Peris. Parameter-independent, curvature-sensitive sprinkle
and star algorithms for surface triangulation. Submitted to the Computer-
Aided Design Journal at Elsevier editorial.
[18] Oscar E. Ruiz and Sebastian Pe˜na. Aspect ratio- and size-controlled
patterned triangulations of parametric surfaces. In Ninth IASTED Intl.
Conf. Computer Graphics and Imaging, February 13-15 2007.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51247", author = "Oscar E. Ruiz and Carlos Cadavid and Juan G. Lalinde and Ricardo Serrano and Guillermo Peris-Fajarnes", title = "Gabriel-constrained Parametric Surface Triangulation", abstract = "The Boundary Representation of a 3D manifold contains
FACES (connected subsets of a parametric surface S : R2 -!
R3). In many science and engineering applications it is cumbersome
and algebraically difficult to deal with the polynomial set and
constraints (LOOPs) representing the FACE. Because of this reason, a
Piecewise Linear (PL) approximation of the FACE is needed, which is
usually represented in terms of triangles (i.e. 2-simplices). Solving the
problem of FACE triangulation requires producing quality triangles
which are: (i) independent of the arguments of S, (ii) sensitive to the
local curvatures, and (iii) compliant with the boundaries of the FACE
and (iv) topologically compatible with the triangles of the neighboring
FACEs. In the existing literature there are no guarantees for the point
(iii). This article contributes to the topic of triangulations conforming
to the boundaries of the FACE by applying the concept of parameterindependent
Gabriel complex, which improves the correctness of the
triangulation regarding aspects (iii) and (iv). In addition, the article
applies the geometric concept of tangent ball to a surface at a point to
address points (i) and (ii). Additional research is needed in algorithms
that (i) take advantage of the concepts presented in the heuristic
algorithm proposed and (ii) can be proved correct.", keywords = "surface triangulation, conforming triangulation, surfacesampling, Gabriel complex.", volume = "2", number = "8", pages = "535-8", }
