Editorial

Marshall Bern
2001 Computational geometry  
Editorial This special issue of Computational Geometry: Theory and Applications is devoted to the topic of combinatorial curves and surfaces. We coined the term combinatorial curves and surfaces not only to denote discrete approximations of continuous shapes, but also to connote an emphasis on combinatorial optimization. Engineers and scientists have been using piecewise-linear meshes for years-what is new is the theoretical analysis of the correctness and efficiency of the meshes. Formulating
more » ... he right notions of correctness and efficiency is itself an important part of the research. The topic of combinatorial curves and surfaces lies at a crossroads of a number of areas, both pure and applied. The applications side includes computer graphics, scientific visualization, geographical information systems, and physical simulation. The pure side includes combinatorial geometry, differential geometry, and geometric and algebraic topology. Part of the attraction of the topic-at least for me personally-is the opportunity to learn tools and ideas from all these different specialties. This issue includes seven papers, giving a snapshot of combinatorial curve and surface research in the year 2000. In "Reconstructing curves with sharp corners", Tamal Dey and Rephael Wenger consider one of the most fundamental problems: reconstructing a curve from a cloud of points in the plane. Most previous algorithms for this problem depend upon the smoothness of the original curve; this paper tackles the more general problem of curves with corners and gives impressive experimental results. The next three papers consider surface reconstruction in two, three and higher dimensions. In "Regular and non-regular point sets: Properties and reconstruction", Sylvain Petitjean and Edmond Boyer define a new notion of what it means to sample a surface sufficiently densely. They offer a definition that requires only a sort of self-consistency, rather than faithful capture of an unknown-and maybe unknowablesmooth surface. In "The power crust, unions of balls, and the medial axis transform", Nina Amenta, Sunghee Choi and Ravi Krishna Kolluri elevate the idea of "polar balls" (from previous work on surface reconstruction by Amenta and myself) to center stage, and build an improved reconstruction algorithm. In particular, the new "power crust" is guaranteed to bound a closed polyhedron. In "Natural neighbor coordinates of points on a surface", Jean-Daniel Boissonnat and Frédéric Cazals generalize Sibson's natural neighbor coordinates to smooth surfaces embedded in any dimension. This generalization induces a higher-order piecewise-polynomial surface reconstruction. The final three papers extend the study of combinatorial curves beyond surface reconstruction. In "Delaunay conforming iso-surface, skeleton extraction and noise removal", Dominique Attali and Jacques-Olivier Lachaud study isosurface extraction and noise removal using the skeleton (medial axis) representation. In "Shape space from deformation", Ho-Lun Cheng, Herbert Edelsbrunner and Ping Fu devise a notion of a "shape space" and describe a program for constructing shape spaces. Shape spaces are built by blending two or more known shapes in continuously changing proportions. Finally, in "Design 0925-7721/01/$ -see front matter 
doi:10.1016/s0925-7721(01)00014-1 fatcat:3edvxqhvmnfs3gpvgkevlj7itu