Computational and Structural Advantages of Circular Boundary Representation

Boundary approximation of planar shapes by circular arcs has quantitive and qualitative advantages compared to using straightline segments. We demonstrate this by way of three basic and frequent computations on shapes - convex hull, decomposition, and medial axis. In particular, we propose a novel medial axis algorithm that beats existing methods in simplicity and practicality, and at the same time guarantees convergence to the medial axis of the original shape.

[1]  Bernard Chazelle,et al.  A theorem on polygon cutting with applications , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[2]  Hazel Everett,et al.  The Graham scan triangulates simple polygons , 1990, Pattern Recognit. Lett..

[3]  Godfried T. Toussaint,et al.  An Efficient Algorithm for Decomposing a Polygon into Star-Shaped Polygons , 1981 .

[4]  Diane Loring Souvaine.  Computational geometry in a curved world (algorithm) , 1986 .

[5]  Dereck S. Meek,et al.  Approximation of a planar cubic Bézier spiral by circular arcs , 1996 .

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

[7]  Jean-Daniel Boissonnat,et al.  Stability and Computation of Medial Axes - a State-of-the-Art Report , 2009, Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration.

[8]  S. Y. Wong,et al.  An optimization approach for biarc curve-fitting of B-spline curves , 1996, Comput. Aided Des..

[9]  Ioannis Z. Emiris,et al.  The Predicates for the Exact Voronoi Diagram of Ellipses under the Euclidiean Metric , 2008, Int. J. Comput. Geom. Appl..

[10]  Wendelin L. F. Degen.  Exploiting curvatures to compute the medial axis for domains with smooth boundary , 2004, Comput. Aided Geom. Des..

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

[12]  D. Walton,et al.  Spiral arc spline approximation to a planar spiral , 1999 .

[13]  Fujio Yamaguchi,et al.  Curves and Surfaces in Computer Aided Geometric Design , 1988, Springer Berlin Heidelberg.

[14]  F. Chazal,et al.  Stability and Finiteness Properties of Medial Axis and Skeleton , 2004 .

[15]  Josef Hoschek,et al.  Handbook of Computer Aided Geometric Design , 2002 .

[16]  Sylvain Pion,et al.  Towards and open curved kernel , 2004, SCG '04.

[17]  F. Chazal,et al.  Stability and homotopy of a subset of the medial axis , 2004, SM '04.

[18]  Tamal K. Dey,et al.  Approximate medial axis as a voronoi subcomplex , 2002, SMA '02.

[19]  David Avis,et al.  A Linear Algorithm for Finding the Convex Hull of a Simple Polygon , 1979, Inf. Process. Lett..

[20]  D. T. Lee,et al.  Location of a point in a planar subdivision and its applications , 1976, STOC '76.

[21]  Martin Held,et al.  Biarc approximation of polygons within asymmetric tolerance bands , 2005, Comput. Aided Des..

[22]  Franz Aurenhammer,et al.  Medial axis computation for planar free-form shapes , 2009, Comput. Aided Des..

[23]  D. Walton,et al.  Approximating smooth planar curves by arc splines , 1995 .

[24]  Hossam A. ElGindy,et al.  A new linear convex hull algorithm for simple polygons , 1984, IEEE Trans. Inf. Theory.

[25]  Astrid Sturm,et al.  Approximation of an open polygonal curve with a minimum number of circular arcs and biarcs , 2008, Comput. Geom..

[26]  Bert Jüttler,et al.  Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics , 2006, Comput. Aided Des..

[27]  Avraham A. Melkman,et al.  On-Line Construction of the Convex Hull of a Simple Polyline , 1987, Inf. Process. Lett..

[28]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[30]  Ronald L. Graham,et al.  An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set , 1972, Inf. Process. Lett..

[31]  Francis Y. L. Chin,et al.  Finding the Medial Axis of a Simple Polygon in Linear Time , 1995, ISAAC.

[32]  J. Koch,et al.  Geometric Hermite interpolation with maximal orderand smoothness , 1996, Comput. Aided Geom. Des..

[33]  Tamal K. Dey,et al.  Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee , 2003, Algorithmica.

[34]  Ulrich Reif.  Uniform B-spline approximation in Sobolev spaces , 2004, Numerical Algorithms.

[35]  Hwan Pyo Moon,et al.  MATHEMATICAL THEORY OF MEDIAL AXIS TRANSFORM , 1997 .

[36]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[37]  L. Kucera,et al.  Randomized incremental construction of abstract Voronoi diagrams , 1993 .

[38]  Kurt Mehlhorn,et al.  Randomized Incremental Construction of Abstract Voronoi Diagrams , 1993, Comput. Geom..

[39]  Jack Sklansky,et al.  Finding the convex hull of a simple polygon , 1982, Pattern Recognit. Lett..

[40]  Universitgtt des Saarlandes.  Fast Triangulation of the Plane with Respect to Simple Polygons , 2005 .

[41]  David P. Dobkin,et al.  Computational geometry in a curved world , 1990, Algorithmica.

[42]  Robert E. Tarjan,et al.  Triangulating a Simple Polygon , 1978, Inf. Process. Lett..

[43]  D. T. Lee,et al.  Medial Axis Transformation of a Planar Shape , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[44]  Ioannis Z. Emiris,et al.  A real-time and exact implementation of the predicates for the Voronoi diagram of parametric ellipses , 2007, Symposium on Solid and Physical Modeling.

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

[46]  Xunnian Yang,et al.  Efficient circular arc interpolation based on active tolerance control , 2002, Comput. Aided Des..