The arc tree: An approximation scheme to represent arbitrary curved shapes

  • Oliver Günther
  • Eugene Wong
Multidimensional Data
Part of the Lecture Notes in Computer Science book series (LNCS, volume 367)


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.


Point Query Curve Point Bezier Curve Abstract Data Type Input Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ball81]
    Ballard, D. H., Strip trees: A hierarchical representation for curves, Comm. of the ACM 24, 5 (May 1981), pages 310–321.Google Scholar
  2. [Bezi74]
    Bezier, P. E., Mathematical and practical possibilities of UNISURF, in Computer Aided Geometric Design, Academic Press, New York, NY, 1974, pages 127–152.Google Scholar
  3. [Bohm84]
    Bohm, W., Efficient evaluation of splines, Computing 33 (1984), pages 171–177.Google Scholar
  4. [Burt77]
    Burton, W., Representation of many-sided polygons and polygonal lines for rapid processing, Comm. of the ACM 20, 3 (March 1977), pages 166–171.Google Scholar
  5. [Debo78]
    Deboor, C., A practical guide to splines, Springer, Heidelberg, 1978.Google Scholar
  6. [Gunt87]
    Gunther, O., An expert database system for the overland search problem, in Proc. BTW'87-Database Systems for Office Automation, Engineering, and Scientific Applications, Informatik-Fachberichte No. 136, Springer, Berlin, 1987.Google Scholar
  7. [Gunt88]
    Gunther, O., Efficient structures for geometric data management, Lecture Notes in Computer Science No. 337, Springer-Verlag, Berlin, 1988.Google Scholar
  8. [Hopc87]
    Hopcroft, J. E. and Krafft, D. B., The challenge of robotics for computer science, in Algorithmic and geometric aspects of robotics, Advances in robotics, Vol. 1, C. Yap and J. Schwartz (eds.), Lawrence Erlbaum Assoc., Hillsdale, NJ, 1987.Google Scholar
  9. [Imai86]
    Imai, H. and Iri, M., Computational-geometric methods for polygonal approximations of a curve, Comp. Vision Graph. Image Proc. 36 (1986), pages 31–41.Google Scholar
  10. [Kemp87]
    Kemper, A., Lockemann, P. C., and Wallrath, M., An object-oriented database system for engineering applications, in Proc. of ACM SIGMOD Conference on Management of Data, San Francisco, Ca., May 1987.Google Scholar
  11. [Kung84]
    Kung, R., Hanson, E., Ioannidis, Y., Sellis, T., Shapiro, L., and Stonebraker, M., Heuristic search in data base systems, in Proc. 1st International Workshop on Expert Database Systems, Kiowah, S.C., Oct. 1984.Google Scholar
  12. [Mand77]
    Mandelbrot, B. B., Fractals: Form, Chance and Dimension, W. H. Freeman & Co., San Francisco, Ca., 1977.Google Scholar
  13. [Meie87]
    Meier, A., Erweiterung relationaler Datenbanksysteme für technische Anwendungen, Informatik-Fachberichte No. 135, Springer, Berlin, 1987.Google Scholar
  14. [Paul87]
    Paul, H.-B., Schek, H.-J., Scholl, M. H., Weikum, G., and Deppisch, U., Architecture and implementation of the Darmstadt database kernel system (DASDBS), in Proc. of ACM SIGMOD Conference on Management of Data, San Francisco, Ca., May 1987.Google Scholar
  15. [Pavl82]
    Pavlidis, T., Algorithms for graphics and image processing, Computer Science Press, Rockville, Md., 1982.Google Scholar
  16. [Ponc87]
    Ponce, J. and Faugeras, O., An object centered hierarchical representation for 3D objects: the prism tree, Comp. Vision Graph. Image Proc. 38 (1987), pages 1–28.Google Scholar
  17. [Prep85]
    Preparata, F. P. and Shamos, M. I., Computational geometry, Springer, New York, NY, 1985.Google Scholar
  18. [RTI84]
    RTI, Relational Technology Inc., INGRES/EQUEL/FORTRAN User's guide, version 3.0, VAX/VMS, Oct. 1984.Google Scholar
  19. [Same84]
    Samet, H., The quadtree and related hierarchical data structures, Computing Surveys 16, 2 (June 1984), pages 187–260.Google Scholar
  20. [Sche86]
    Schek, H.-J., Datenbanksysteme für die Verwaltung geometrischer Objekte, in Proc. of the 16th GI Annual Meeting, Informatik-Fachberichte No. 126, Springer, Berlin, Oct. 1986.Google Scholar
  21. [Ston83]
    Stonebraker, M., Rubenstein, B., and Guttman, A., Application of abstract data types and abstract indices to CAD data, in Proc. Engineering Applications Stream of ACM SIGMOD Conference, San Jose, Ca., May 1983.Google Scholar
  22. [Ston86a]
    Stonebraker, M. and Rowe, L., The design of POSTGRES, in Proc. of ACM SIGMOD Conference on Management of Data, Washington, DC, June 1986.Google Scholar
  23. [Ston86b]
    Stonebraker, M., Object management in POSTGRES using procedures, in Proc. 1986 International Workshop on Object-Oriented Database Systems, Asilomar, Ca., Sept. 1986.Google Scholar
  24. [Wong85]
    Wong, E., Extended domain types and specification of user defined operators, U.C. Berkeley, Memorandum No. UCB/ERL/M85/3, Feb. 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Oliver Günther
    • 1
  • Eugene Wong
    • 2
  1. 1.Department of Computer ScienceUniversity of CaliforniaSanta Barbara
  2. 2.Department of EECSUniversity of CaliforniaBerkeley

Personalised recommendations