UNIT 32 - SIMPLE ALGORITHMS I - INTERSECTION OF LINES

UNIT 32 - SIMPLE ALGORITHMS I - INTERSECTION OF LINES

Compiled with assistance from David H. Douglas, University of Ottawa and David M. Mark, State University of New York at Buffalo

  • A. INTRODUCTION
  • B. DEFINITIONS
  • C. SIMPLEST CASE
  • D. SPECIAL CASES
  • E. COMPLEX LINES
  • REFERENCES
  • DISCUSSION AND EXAM QUESTIONS
  • NOTES

    UNIT 32 - SIMPLE ALGORITHMS I - INTERSECTION OF LINES

    Compiled with assistance from David H. Douglas, University of Ottawa and David M. Mark, State University of New York at Buffalo

    A. INTRODUCTION

    B. DEFINITIONS

    Algorithms

    Heuristics

    C. SIMPLEST CASE

    Question

    Procedure

    Solution

    General form

    Simple program

    D. SPECIAL CASES

    Vertical lines

    Parallel lines

    Solution

    E. COMPLEX LINES

    Minimum enclosing rectangle

    Monotonic sections

    Sorting lines

    REFERENCES

    Douglas, David H., 1974. "It makes me so cross," a paper distributed by the Laboratory for Computer Graphics and Spatial Analysis, Graduate School of Design, Harvard University, September 1974. (This note has been reprinted numerous times in computing magazines, and in Marble, Calkins and Peuquet, 1984.)

    Lee, D.T. and F.P. Preparata, 1984. "Computational geometry: a survey," IEEE Transactions on Computers C-33(12):1072- 1101. A good introduction to basic algorithms for geometrical problems.

    Little, J.J. and T.K. Peucker, 1979. "A recursive procedure for finding the intersection of two digital curves," Computer Graphics and Image Processing 10:159-71.

    Marble, Duane F., Calkins, Hugh W. and Peuquet, Donna J., eds., 1984. Basic Readings in Geographic Information Systems, Williamsville N.Y., SPAD Systems Ltd.,

    Saalfeld, Alan, 1987. "It doesn't make me nearly as CROSS," International Journal of Geographical Information Systems 1(4), pp 379-386

    Taylor, George, 1989. "Letters," International Journal of Geographical Information Systems 3(20):192-3. Another look at the line intersection problem.

    DISCUSSION AND EXAM QUESTIONS

    1. Why is the "crossing segment" or line intersection problem so important in GIS?

    2. Identify the rules which can be used to limit searching for intersections between two lines which are both monotonic in x and y. Each line will be one of four combinations - either increasing or decreasing in x, and either increasing or decreasing in y. You will need to deal with 16 combinations when discussing the options for two lines.

    3. Review and discuss the technique for line intersection described in Little and Peucker, 1979.

    4. Compare raster and vector approaches to the determination of the intersection between two lines. Are there circumstances under which a raster approach might be preferable?

    5. Since geographical data is never perfectly accurate, the special cases identified in the algorithm should never occur precisely. Modify the algorithm to deal with special cases by treating data as imprecise. What are the advantages and potential problems with such an approach?


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