Computational and Structural Advantages of Circular Boundary Representation

Boundary approximation of planar shapes by circular arcs has quantitative and qualitative advantages compared to using straight-line 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]  Godfried T. Toussaint,et al.  An Efficient Algorithm for Decomposing a Polygon into Star-Shaped Polygons , 1981 .

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

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

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

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

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

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

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

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

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

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

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

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

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

[15]  F. Frances Yao,et al.  Finding the Convex Hull of a Simple Polygon , 1983, J. Algorithms.

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

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

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

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

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

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

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

[23]  Franco P. Preparata,et al.  Location of a Point in a Planar Subdivision and Its Applications , 1977, SIAM J. Comput..

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

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

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

[27]  Kurt Mehlhorn,et al.  Fast Triangulation of the Plane with Respect to Simple Polygons , 1985, Inf. Control..

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

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

[30]  Otfried Cheong,et al.  The Voronoi Diagram of Curved Objects , 2005, Discret. Comput. Geom..

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

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

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

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