Precise Voronoi Cell Extraction of Free-Form Planar Piecewise C1-Continuous Closed Rational Curves

We present an algorithm for generating Voronoi cells for a set of planar piecewise C1-continuous closed rational curves, which is precise up to machine precision. The algorithm starts with the symbolically generated bisectors for pairs of C1-continuous curve segments (C(t),Ci(r)). The bisectors are represented implicitly in the tr-parameter space. Then, they are properly trimmed after being split into monotone pieces. The trimming procedure uses the orientation of the original curves as well as their curvature fields, resulting in a set of trimmed-bisector segments represented as implicit curves in a parameter space. A lower-envelope algorithm is then used in the parameter space of the curve whose Voronoi cell is sought. The lower envelope represents the exact boundary of the Voronoi cell. The algorithm also supports piecewise C1-continuous curves and generates the Voronoi cell of such input curves using additional point/curve bisector segments.

[1]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[2]  J. O´Rourke,et al.  Computational Geometry in C: Arrangements , 1998 .

[3]  Jin J. Chou.  Voronoi diagrams for planar shapes , 1995, IEEE Computer Graphics and Applications.

[4]  B. Gurumoorthy,et al.  Constructing medial axis transform of planar domains with curved boundaries , 2003, Comput. Aided Des..

[5]  James H. Davenport,et al.  Voronoi diagrams of set-theoretic solid models , 1992, IEEE Computer Graphics and Applications.

[6]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[7]  R. Brubaker.  Models for the perception of speech and visual form: Weiant Wathen-Dunn, ed.: Cambridge, Mass., The M.I.T. Press, I–X, 470 pages , 1968 .

[8]  Gershon Elber,et al.  Bisector curves of planar rational curves , 1998, Comput. Aided Des..

[9]  R. Farouki,et al.  Voronoi diagram and medial axis algorithm for planar domains with curved boundaries I. Theoretical foundations , 1999 .

[10]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[11]  Gershon Elber,et al.  Geometric constraint solver using multivariate rational spline functions , 2001, SMA '01.

[12]  Christoph M. Hoffmann,et al.  Eliminating extraneous solutions in curve and surface operations , 1991, Int. J. Comput. Geom. Appl..

[13]  Rida T. Farouki,et al.  Specified-Precision Computation of Curve/Curve Bisectors , 1998, Int. J. Comput. Geom. Appl..

[14]  Cecil G. Armstrong,et al.  Modelling requirements for finite-element analysis , 1994, Comput. Aided Des..

[15]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[16]  H. N. Gürsoy,et al.  An automatic coarse and fine surface mesh generation scheme based on medial axis transform: Part i algorithms , 1992, Engineering with Computers.

[17]  Otfried Cheong,et al.  The Voronoi Diagram of Curved Objects , 1995, SCG '95.

[18]  Gabriella Sanniti di Baja,et al.  (3, 4)-weighted Skeleton Decomposition for Pattern Representation and Description , 1994, Pattern Recognit..

[19]  Martin Held,et al.  Voronoi diagrams and offset curves of curvilinear polygons , 1998, Comput. Aided Des..

[20]  Vijay Srinivasan,et al.  Voronoi Diagram for Multiply-Connected Polygonal Domains I: Algorithm , 1987, IBM J. Res. Dev..

[21]  H. Blum.  Biological shape and visual science (part I) , 1973 .

[22]  Chee-Keng Yap,et al.  AnO(n logn) algorithm for the voronoi diagram of a set of simple curve segments , 1987, Discret. Comput. Geom..

[23]  Deok-Soo Kim,et al.  Representing the Voronoi diagram of a simple polygon using rational quadratic Bézier curves , 1995, Comput. Aided Des..

[24]  Journal of the Association for Computing Machinery , 1961, Nature.

[25]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[26]  Ju-Hsien Kao,et al.  Process planning for additive/subtractive solid freeform fabrication using medial axis transform , 1999 .

[27]  Chee-Keng Yap,et al.  Towards Exact Geometric Computation , 1997, Comput. Geom..

[28]  Dinesh Manocha,et al.  Efficient and exact manipulation of algebraic points and curves , 2000, Comput. Aided Des..

[29]  R. Farouki,et al.  The bisector of a point and a plane parametric curve , 1994, Comput. Aided Geom. Des..

[30]  Gershon Elber,et al.  Precise Voronoi cell extraction of free-form rational planar closed curves , 2005, SPM '05.

[31]  Gershon Elber,et al.  The bisector surface of rational space curves , 1998, TOGS.

[32]  Ugo Montanari,et al.  Continuous Skeletons from Digitized Images , 1969, JACM.