hw4-sol - AMS 345/CSE 355 (Spring, 2009) Joe Mitchell...

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Unformatted text preview: AMS 345/CSE 355 (Spring, 2009) Joe Mitchell COMPUTATIONAL GEOMETRY Homework Set # 4 – Solution Notes (1). [20 points] O’Rourke, problem 4, section 3.2.3, page 68. In two dimensions, the affine hull of two points is the line through the two points, of three noncolinear points is the plane ℜ 2 , of n > 3 points not all on the same line is also the plane ℜ 2 . The affine hull of n ≥ 3 points that are collinear is the line through the points. In three dimensions, the affine hull of two points is the line through the two points, of three noncolinear points is the plane through the three points, of four points in general position is all of space, ℜ 3 , of n > 4 points in general position (or at least some subset of four of them in general position) is all of space, ℜ 3 . The affine hull of n ≥ 3 points that are collinear is the line through the points. The affine hull of n ≥ 4 points that are coplanar is the plane through the points. (2). [15 points] O’Rourke, problem 3, section 3.5.7, page 86. In the figure on the right below, we illustrate the case in which the points indexed 0 through n − 2 are in convex position, but the point indexed n − 1 is positioned so that the convex hull of all n points is a triangle, (0,1, n − 1). Then, Algorithm 3.6 will enter the while loop with i = 2 and push points 2, 3, . . . , n − 2, each time incrementing i by 1. Once i = n − 1, when we enter the while loop next, we pop n − 2 (no change to i ), then pop n − 3 (no change to i ), etc, until we pop point indexed by 2 (and i = n − 1 still). In the final iteration of the while loop, we push point n − 1, set i = n , and the while loop has ended (since i < n is false). In total there were n − 3 iterations of the while loop that do a push, then n − 2 iterations that do a pop, then 1 iteration that pushes; in total, there are 2 n − 5 iterations. Note that the same will happen in any case in which the convex hull has only 3 vertices ( h = 3), even of the points that are interior to the hull are not in convex position (as in the figure below): all but 3 points will be both pushed and popped. 1 2 3 n-1 n-1 n-2 1 2 3 (3). [20 points] Let S be the set of points { (3,1), (4,4), (5,-1), (2,-1), (2,2), (2,1), (4,-1), (6,1), (8,1), (8,5), (7,4), (3,3) } ....
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This note was uploaded on 02/07/2010 for the course AMS 345 taught by Professor Mitchell,j during the Spring '08 term at SUNY Stony Brook.

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hw4-sol - AMS 345/CSE 355 (Spring, 2009) Joe Mitchell...

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