Abstract
This chapter discusses algorithm techniques related to geometry. The general goal of the chapter is to find ways to conveniently solve geometric problems, avoiding special cases and tricky implementations. Section 13.1 introduces the C++ complex number class which has useful tools for geometric problems. After this, we will learn to use cross products to solve various problems, such as testing whether two line segments intersect and calculating the distance from a point to a line. Finally, we discuss ways to calculate polygon areas and explore special properties of Manhattan distances. Section 13.2 focuses on sweep line algorithms which play an important role in computational geometry. We will see how to use such algorithms for counting intersection points, finding closest points, and constructing convex hulls.
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Notes
- 1.
Creating an efficient algorithm for the closest pair problem was once an important open problem in computational geometry. Finally, Shamos and Hoey [26] discovered a divide and conquer algorithm that works in \(O(n \log n)\) time. The sweep line algorithm presented here has common elements with their algorithm, but it is easier to implement.
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Laaksonen, A. (2017). Geometry. In: Guide to Competitive Programming. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-72547-5_13
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DOI: https://doi.org/10.1007/978-3-319-72547-5_13
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