Approximating parametric curves with strip trees using affine arithmetic

This paper introduces the arc tree, a hierarchical data structure to represent arbitrary curved shapes. The arc tree is a balanced binary tree that represents a curve of length l such that any subtree whose root is on the k-th tree level is representing a subcurve of length l/2 k . Each tree level is associated with an approximation of the curve; lower levels correspond to approximations of higher resolution. Based on this hierarchy of detail, queries such as point search or intersection detection and computation can be solved in a hierarchical manner. We compare the arc tree to several related schemes and present the results of a practical performance analysis for various kinds of set and search operators. We also discuss several options to embed arc trees as complex objects in an extensible database management system and argue that the embedding as an abstract data type is most promising.

[1]  P. Bézier MATHEMATICAL AND PRACTICAL POSSIBILITIES OF UNISURF , 1974 .

[2]  Warren Burton,et al.  Representation of many-sided polygons and polygonal lines for rapid processing , 1977, CACM.

[3]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[4]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[5]  I. Faux,et al.  Computational Geometry for Design and Manufacture , 1979 .

[6]  Fred D. Crary A Versatile Precompiler for Nonstandard Arithmetics , 1979, TOMS.

[7]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[8]  Dana H. Ballard,et al.  Strip trees: a hierarchical representation for curves , 1981, CACM.

[9]  Carlo Zaniolo,et al.  The database language GEM , 1983, SIGMOD '83.

[10]  Michael Stonebraker,et al.  Application of Abstract Data Types and Abstract Indices to CAD Data Bases , 1986, Engineering Design Applications.

[11]  G. Toussaint Solving geometric problems with the rotating calipers , 1983 .

[12]  R.M. McElhaney,et al.  Algorithms for graphics and image processing , 1983, Proceedings of the IEEE.

[13]  Theo Pavlidis,et al.  Algorithms for Graphics and Imag , 1983 .

[14]  Hanan Samet,et al.  The Quadtree and Related Hierarchical Data Structures , 1984, CSUR.

[15]  Michael Stonebraker,et al.  Heuristic Search in Data Base Systems , 1984, Expert Database Workshop.

[16]  Michael Stonebraker,et al.  Object Management in Postgres using Procedures , 1986, On Object-Oriented Database System.

[17]  Hiroshi Imai,et al.  Computational-geometric methods for polygonal approximations of a curve , 1986, Comput. Vis. Graph. Image Process..

[18]  Michael Stonebraker,et al.  The design of POSTGRES , 1986, SIGMOD '86.

[19]  Andreas Meier,et al.  Erweiterung relationaler Datenbanksysteme für technische Anwendungen , 1987, Informatik-Fachberichte.

[20]  Chee-Keng Yap,et al.  Algorithmic and geometric aspects of robotics , 1987 .

[21]  Hans-Jörg Schek,et al.  Architecture and implementation of the Darmstadt database kernel system , 1987, SIGMOD '87.

[22]  Alan H. Barr,et al.  Accurate triangulations of deformed, intersecting surfaces , 1987, SIGGRAPH.

[23]  A. Kemper,et al.  An Object-Oriented Database System for Engineering Applications , 1987, SIGMOD Conference.

[24]  Olivier D. Faugeras,et al.  An object centered hierarchical representation for 3D objects: The prism tree , 1987, Comput. Vis. Graph. Image Process..

[25]  Peter C. Lockemann,et al.  An object-oriented system for engineering applications , 1987, SIGMOD '87.

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

[27]  Oliver Günther,et al.  An Expert Database System For The Overland Search Problem , 1987, BTW.

[28]  Oliver Günther,et al.  Efficient Structures for Geometric Data Management , 1988, Lecture Notes in Computer Science.

[29]  Hanan Samet,et al.  The Design and Analysis of Spatial Data Structures , 1989 .

[30]  Eugene Wong,et al.  The arc tree: An approximation scheme to represent arbitrary curved shapes , 1990, Comput. Vis. Graph. Image Process..

[31]  J. Stolfi,et al.  Aane Arithmetic and Its Applications to Computer Graphics , 1990 .

[32]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.