Multiple line segment intersection

In computational geometry, the line segment intersection problem supplies a list of line segments in the plane and asks us to determine whether any two of them intersect, or cross.

Naive algorithms examine each pair of segments, but for a high number of possibly intersecting segments this becomes increasingly inefficient since most pairs of segments aren't anywhere close to one another in a typical input sequence.

The most common, more efficient way to solve this problem for a high number of segments is to use a sweep line algorithm, where we imagine a line sliding across the line segments and we track which line segments it intersects at each point in time using a dynamic data structure based on binary search trees. The Shamos–Hoey algorithm applies this principle to solve the line segment intersection detection problem, as stated above, of determining whether or not a set of line segments has an intersection; the Bentley–Ottmann algorithm works by the same principle to list all intersections in logarithmic time per intersection.

See also

• Line-line intersection

References

• Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf (2000). Computational Geometry (2nd edition ed.). Springer. ISBN 3-540-65620-0. {{cite book}}: |edition= has extra text (help)CS1 maint: multiple names: authors list (link) Chapter 2: Line Segment Intersection, pp.19–44.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 1990. ISBN 0-262-03293-7. Section 33.2: Determining whether any pair of segments intersects, pp.934–947.
• J. L. Bentley and T. Ottmann., Algorithms for reporting and counting geometric intersections, IEEE Trans. Comput. C28 (1979), 643–647.

External links

• http://softsurfer.com/Archive/algorithm_0108/algorithm_0108.htm#Pseudo-Code:%20B-O
• Robert Pless. Lecture 4 notes. Washington University in St. Louis, CS 506: Computational Geometry.
• Line segment intersection in CGAL, the Computational Geometry Algorithms Library
• For the simple case of testing the intersection of two line segments: Solution by Paul Bourke.
• "Line Segment Intersection" lecture notes by Jeff Erickson.
• Line-Line Intersection Method With C Code Sample Darel Rex Finley