Reference no: EM13908722
1. Given points P1 = (1/2, 1/2), P2 = (1,0) and P3 = (2,2), we will use the duality that maps P = (x, y) to the line TP = {(u, v): v = xu + y and the line l with equation y = mx + b maps to the point Tl = (-m,b).
(a) Carefully describe the dual of the triangle ΔP1P2P3
(b) Repeat, now using the polar transformation for duality.
(c) Describe the (usual) dual of n points, P1,......,P2 in convex position in R2. Repeat if n = 3k and the set is is general position with k convex layers".
L = {l1,...,ln} is a set of n lines in the plane in general position.
2. Given i,j ∈ (1,...,n) and L as above, you want to decide if li ∈ λi; that is if the ith line of L meets the jth level of A(L).
(a) First show that if li has the kth smallest slope, the answer is YES if j is between k and n - k + 1.
(b) Now give an efficient algorithm in case the property in (a), above, does not hold.
(c) (*) Try to give a nontrivial lower bound for this task.
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