Carefully describe the dual of the triangle - repeat now

Question

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,...,l_n} 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 l_i ∈ λ_i; that is if the i^th line of L meets the j^th level of A(L).

(a) First show that if l_i has the k^th 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.