Module8_08 - Computational Geometry Further Reading TOPICS...

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CSE5311 1 Computational Geometry TOPICS Preliminaries Point in a Polygon Polygon Construction Convex Hulls Further Reading

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CSE5311 2 Geometric Algorithms Geometric Algorithms find applications in such areas as Computer Graphics Computer Aided Design VLSI Design GIS Robotics We will study algorithms dealing with points, lines, line segments, and polygons In particular, the algorithms will Determine whether a point is inside a Polygon Construct a Polygon Determine Convex Hulls
CSE5311 3 Preliminaries: A point p is represented as a pair of coordinates (x,y) A line is represented by a pair of points A path is a sequence of points p 1 ,p 2 , . . . p n and the line segments connecting them, p 1 -p 2 , p 2 -p 3 , . . . , p k-1 -p k . A closed path whose last point is the same as the first is a polygon. A simple polygon is one whose corresponding path does not intersect itself. It encloses a region in the plane. A convex Polygon is a polygon such that any line segment connecting two points inside the polygon is itself entirely in the polygon. The convex hull of a set of points is defined as the smallest convex polygon enclosing all the given points.

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CSE5311 4 A line segment connecting two points: The points are inside the polygon The line segment is not entirely in the polygon This is not a convex polygon p(x 1 ,y 1 ) q(x 2 ,y 2 ) r(x 3 ,y 3 ) s(x 4 ,y 4 )
CSE5311 5 Determining whether a point is inside a polygon Given a simple polygon polygon P, and a point q, determine whether the point is inside or outside the polygon. (a non-convex polygon)

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CSE5311 6 Procedure Point_in_a_Polygon(P,q) Input : P ( a simple polygon with vertices p 1 ,p 2 ,p 3 , and edges e 1 ,e 2 ,e 3 , …e n and q (x 0 ,y 0 ) a point. Output:
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Module8_08 - Computational Geometry Further Reading TOPICS...

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