( x + pA + qB, y + qA − pB ) for all A, B ∈ Z 16 th Q ( s ) � E � = Q ( s ) � e 1 , . . . , e k � We have Every element of Q ( s ) International Symposium δ i � x � = 0 y i = � j y ij e j for all y i ∈ Y on Graph can be written as p/q , δ i y = 0 Drawing Moreover from what we wrote above we have: where p, q ∈ O Q ( s ) . 0 ≤ l ≤ h x ijl s l and y ij = � 0 ≤ l ≤ h y ijl s l x ij = � � δ i x = pA i + qB i � = 0 GCD is 1. ∆ x = � i,j ∆ x ij e j rizontal, ∆ y i = sa i − b i = 0 , δ i y = qA i − pB i = 0 x ij = a ij /b ij Connected D x = { d x ∈ R : d x = | x ( v lft ) − x ( v ) | , ∀ v ∈ V } a i + sb i = a i + s 2 a i = (1 + s 2 ) a i . y ij = c ij /d ij Rectilinear Graphs on for all odd values of i , 0 < i < n and Point Sets We consider the following cases: ertical, then ∆ x i = a i + sb i = 0 , q ( p 2 + q 2 ) y = A i δ i 1. [ Q ( s ) : Q ] = 1 , s is rational. � δ i sa i − b i = − s 2 b i − b i = − (1+ s 2 ) b i . x = pA i + qB i = 0 δ i y = qA i − pB i � = 0 2. [ Q ( s ) : Q ] < ∞ , s is algebraic over Q . x i , y i ∈ O Q ( s ) � E � 3. [ Q ( s ) : Q ] = ∞ , s is transcendental over Q . 0 ≤ l ≤ h w l s l Maarten L¨ offler � for all even values of i , 0 < i < n . Utrecht University w = 0 ≤ l ≤ h x ijl s l Elena Mumford In total we move from v i to v i +1 x ij = � 0 ≤ l ′≤ h ′ w ′ l ′ s l ′ Technical University Eindhoven over a distance ( a i + sb i , sa i − b i ) � where a i , b i ∈ Z � E � . 0 ≤ l ≤ h y ijl s l the Netherlands y ij = � 1-1
( x + pA + qB, y + qA − pB ) for all A, B ∈ Z 16 th Q ( s ) � E � = Q ( s ) � e 1 , . . . , e k � We have Every element of Q ( s ) International Symposium δ i � x � = 0 y i = � j y ij e j for all y i ∈ Y on Graph can be written as p/q , δ i y = 0 Drawing Moreover from what we wrote above we have: where p, q ∈ O Q ( s ) . 0 ≤ l ≤ h x ijl s l and y ij = � 0 ≤ l ≤ h y ijl s l x ij = � � δ i x = pA i + qB i � = 0 GCD is 1. ∆ x = � i,j ∆ x ij e j rizontal, ∆ y i = sa i − b i = 0 , δ i y = qA i − pB i = 0 x ij = a ij /b ij Connected D x = { d x ∈ R : d x = | x ( v lft ) − x ( v ) | , ∀ v ∈ V } a i + sb i = a i + s 2 a i = (1 + s 2 ) a i . y ij = c ij /d ij Rectilinear Graphs on for all odd values of i , 0 < i < n and Point Sets We consider the following cases: ertical, then ∆ x i = a i + sb i = 0 , q ( p 2 + q 2 ) y = A i δ i 1. [ Q ( s ) : Q ] = 1 , s is rational. � δ i sa i − b i = − s 2 b i − b i = − (1+ s 2 ) b i . x = pA i + qB i = 0 δ i y = qA i − pB i � = 0 2. [ Q ( s ) : Q ] < ∞ , s is algebraic over Q . x i , y i ∈ O Q ( s ) � E � 3. [ Q ( s ) : Q ] = ∞ , s is transcendental over Q . 0 ≤ l ≤ h w l s l Maarten L¨ offler � for all even values of i , 0 < i < n . Utrecht University w = 0 ≤ l ≤ h x ijl s l Elena Mumford In total we move from v i to v i +1 x ij = � 0 ≤ l ′≤ h ′ w ′ l ′ s l ′ Technical University Eindhoven over a distance ( a i + sb i , sa i − b i ) � where a i , b i ∈ Z � E � . 0 ≤ l ≤ h y ijl s l the Netherlands y ij = � 1-2
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Question: How many orientations can a point set have, such that the maximal axis-parallel graph is connected? 4-1
Question: How many orientations can a point set have, such that the maximal axis-parallel graph is connected? [Therese Biedl, 2007] 4-2
Let’s look at an example point set in all possible orientation. 5-1
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Answer: One. 8-1
Answer: One. (up to trivial rotations) 8-2
Start simple: two dimensions, integer coordinates, rotation over 45 ◦ . 9-1
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That’s good, but what if my coordinates are not integers? 16-1
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What about other slopes than 45 ◦ ? 21-1
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So, what happens in higher dimensions? 25-1
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Any Questions... ? 27-1
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