Abstract
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.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
Ballard, D. H., Strip trees: A hierarchical representation for curves, Comm. of the ACM 24, 5 (May 1981), pages 310–321.
Bezier, P. E., Mathematical and practical possibilities of UNISURF, in Computer Aided Geometric Design, Academic Press, New York, NY, 1974, pages 127–152.
Bohm, W., Efficient evaluation of splines, Computing 33 (1984), pages 171–177.
Burton, W., Representation of many-sided polygons and polygonal lines for rapid processing, Comm. of the ACM 20, 3 (March 1977), pages 166–171.
Deboor, C., A practical guide to splines, Springer, Heidelberg, 1978.
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.
Gunther, O., Efficient structures for geometric data management, Lecture Notes in Computer Science No. 337, Springer-Verlag, Berlin, 1988.
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.
Imai, H. and Iri, M., Computational-geometric methods for polygonal approximations of a curve, Comp. Vision Graph. Image Proc. 36 (1986), pages 31–41.
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.
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.
Mandelbrot, B. B., Fractals: Form, Chance and Dimension, W. H. Freeman & Co., San Francisco, Ca., 1977.
Meier, A., Erweiterung relationaler Datenbanksysteme für technische Anwendungen, Informatik-Fachberichte No. 135, Springer, Berlin, 1987.
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.
Pavlidis, T., Algorithms for graphics and image processing, Computer Science Press, Rockville, Md., 1982.
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.
Preparata, F. P. and Shamos, M. I., Computational geometry, Springer, New York, NY, 1985.
RTI, Relational Technology Inc., INGRES/EQUEL/FORTRAN User's guide, version 3.0, VAX/VMS, Oct. 1984.
Samet, H., The quadtree and related hierarchical data structures, Computing Surveys 16, 2 (June 1984), pages 187–260.
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.
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.
Stonebraker, M. and Rowe, L., The design of POSTGRES, in Proc. of ACM SIGMOD Conference on Management of Data, Washington, DC, June 1986.
Stonebraker, M., Object management in POSTGRES using procedures, in Proc. 1986 International Workshop on Object-Oriented Database Systems, Asilomar, Ca., Sept. 1986.
Wong, E., Extended domain types and specification of user defined operators, U.C. Berkeley, Memorandum No. UCB/ERL/M85/3, Feb. 1985.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Günther, O., Wong, E. (1989). The arc tree: An approximation scheme to represent arbitrary curved shapes. In: Litwin, W., Schek, HJ. (eds) Foundations of Data Organization and Algorithms. FODO 1989. Lecture Notes in Computer Science, vol 367. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51295-0_142
Download citation
DOI: https://doi.org/10.1007/3-540-51295-0_142
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51295-0
Online ISBN: 978-3-540-46186-9
eBook Packages: Springer Book Archive