Algorithms , Probability , and Computing Solutions KW 41 HS 17 Solution of In - Class Exercise 1 : Lines Intersecting Segment

Solution of In-Class Exercise 1: Lines Intersecting Segment Let s be a line segment in the plane R, and let u, v 2 R be the two endpoints that delimit it. Let furthermore L be the set of all non-vertical lines l that intersect s. A simple characterization of L would be that any non-veritcal line l is contained in L if and only if one of the following occurs: (i) l contains u or (ii) l contains v or (iii) u lies above l and v lies below l or (iv) v lies above l and u lies below l. In the dual space, u and v are two xed lines and we are looking for the set of points L corresponding to the duals of all lines in L. According to Observation 2.3 and the preceding discussion, the above criteria for l hitting the segment s are satis ed if and only if in the dual, (i') l lies on u (incidence) or (ii') l lies on v (incidence) or (iii') l lies above u and l lies below v (2.3) or (iv') l lies above v and l lies below u (2.3) So the space L we are looking for consists of four regions de ned by these criteria. The following picture shows what those regions typically look like. On the left, the primal space is shown, containing a line segment s and several lines f, g, h intersecting s in exactly one point, where on the right their duals are shown. Here `s is the line going through the endpoints u and v. The set of points L consists of the two lines u and v as well as all points in the left wedge that have u above and v below (region (iv')) and all points in the right wedge that have v above and u below (region (iii')).