Maximizing Angles in Plane Straight Line Graphs Oswin Aichholzer, TU Graz Thomas Hackl, TU Graz Michael Hoffmann, ETH Zürich Clemens Huemer, UP Catalunya Attila Pór, Charles U Francisco Santos, U de Cantabria Bettina Speckmann, TU Eindhoven Birgit Vogtenhuber, TU Graz

s.t. rotation needed is minimal. Optimal Surveillance Place a rotating camera to observe all edges

s.t. it leaves out the maximum incident angle. Optimal Surveillance Place a rotating camera to observe all edges s.t. rotation needed is minimal.

Optimal Surveillance Connect a set of points, s.t. at each point there is a large incident angle. n P ½ R 2

Optimal Surveillance On any set of points there is a graph, s.t. at each vertex there is a large incident angle.

Openness of a PSLG -open 2 ¼ = 3 A is -open iff each vertex has an incident angle of size. ¸ ' ' PSLG

Triangulations Wlog. CH is a triangle. For each finite point set in general position there exists a –open triangulation. 2 ¼ = 3

Triangulations marked angles ∑ = 8 ¼ one light angle ¸ 2 ¼ = 3 pick point and recurse… For each finite point set in general position there exists a –open triangulation. 2 ¼ = 3 light angles ∑ ¸ 2 ¼

Triangulations … … For each finite point set in general position there exists a –open triangulation. 2 ¼ = 3 one light angle ¸ 2 ¼ = 3 pick point and recurse… marked angles ∑ = 8 ¼ light angles ∑ ¸ 2 ¼

Spanning Trees (a,b) diameter a b c O1. Any angle opposite to a diameter is bad. O2. In any triangle at least one angle is good. good angle ≤ ¼ = 3 For each finite point set in general position there exists a –open spanning tree. 5 ¼ = 3 ? ? bad angle = 3 ¼ = 3 >

Spanning Trees a b d c c,d in max. distance to (a,b) wlog For each finite point set in general position there exists a –open spanning tree. 5 ¼ = 3 (a,b) diameter good angle ≤ ¼ = 3 bad angle = 3 ¼ = 3 >

Spanning Trees a b c d c,d in max. distance to (a,b) supp. For each finite point set in general position there exists a –open spanning tree. 5 ¼ = 3 (a,b) diameter good angle ≤ ¼ = 3 bad angle = 3 ¼ = 3 >

Spanning Trees a b d c c,d in max. distance to (a,b) supp. For each finite point set in general position there exists a –open spanning tree. 5 ¼ = 3 (a,b) diameter good angle ≤ ¼ = 3 bad angle = 3 ¼ = 3 >

Spanning Trees a b d c c,d in max. distance to (a,b) supp. For each finite point set in general position there exists a –open spanning tree. 5 ¼ = 3 (a,b) diameter good angle ≤ ¼ = 3 bad angle = 3 ¼ = 3 >

Spanning Trees a b d c c,d in max. distance to (a,b) wlog For each finite point set in general position there exists a –open spanning tree. 5 ¼ = 3 (a,b) diameter good angle ≤ ¼ = 3 bad angle = 3 ¼ = 3 >

; Spanning Trees a b c d c,d in max. distance to (a,b) supp. For each finite point set in general position there exists a –open spanning tree. 5 ¼ = 3 (a,b) diameter good angle ≤ ¼ = 3 bad angle = 3 ¼ = 3 >

; Spanning Trees a b c,d in max. distance to (a,b) e d For each finite point set in general position there exists a –open spanning tree. 5 ¼ = 3 (a,b) diameter good angle ≤ ¼ = 3 bad angle = 3 ¼ = 3 > c

Recap: Results For any finite point set in general position … there exists a –open spanning tree. 5 ¼ = 3 there exists a –open triangulation. 2 ¼ = 3

… Best possible even for degree at most n-2. For any finite point set in general position there exists a -open spanning tree of maximum vertex degree three. 3 ¼ = 2 Spanning Trees with Δ ≤ 3

(a,b) diameter a b B A and bridge in the tree. OBS: angles at a and b are ok. For any finite point set in general position there exists a -open spanning tree of maximum vertex degree three. 3 ¼ = 2

Spanning Trees with Δ ≤ 3 (c,d) diameter of A a b Continue recursively  max degree 4 c d ? ? DC- C+ For any finite point set in general position there exists a -open spanning tree of maximum vertex degree three. 3 ¼ = 2

Spanning Trees with Δ ≤ 3 (c,d) diameter of A a b Continue recursively  max degree 4 c d DC- C+ One of C+ or C- is empty  c has degree 3 C For any finite point set in general position there exists a -open spanning tree of maximum vertex degree three. 3 ¼ = 2

Spanning Trees with Δ ≤ 3 (c,d) diameter of A a b Consider tangents from a to C. c d D Only one set per vertex  maxdegree 3. C2 C1C3 For any finite point set in general position there exists a -open spanning tree of maximum vertex degree three. 3 ¼ = 2

Spanning Paths for Convex Sets For any finite point set P in convex position there exists a –open spanning path. 3 ¼ = 2 Zig-zag paths# = n At most one bad zig-zag angle per vertex. No bad zig-zag angle at diametrical vertices.  At least two good zig-zag paths.

Spanning Paths For any finite point set P in general position there exists a –open spanning path. 5 ¼ = 4 1) For any finite point set P in general position and each vertex q of its convex hull there exists a qqq–open spanning path with endpoint q. 5 ¼ = 4 2) For any finite point set P in general position and each edge q 1 q 2 of its convex hull there exists a qqqqqq–open spanning path (q 1,q 2,…) or (q 2,q 1,…). 5 ¼ = 4

CH(P\{q}) Spanning Paths For each finite point set in general position there exists a –open spanning path. 5 ¼ = 4 q z y 1) q vertex of CH bad angle ¸ 3¼ = 4 good angle · 3¼ = 4 a) q in normal cone of an edge yz

CH(P\{q}) Spanning Paths For each finite point set in general position there exists a –open spanning path. 5 ¼ = 4 q 1) q vertex of CH bad angle ¸ 3¼ = 4 good angle · 3¼ = 4 b) q in normal cone of a vertex p i) Angle zpy is good one is < ¼ z y p

CH(P\{q}) Spanning Paths For each finite point set in general position there exists a –open spanning path. 5 ¼ = 4 q 1) q vertex of CH bad angle ¸ 3¼ = 4 good angle · 3¼ = 4 b) q in normal cone of a vertex p i) Angle ypq is good (wlog) z y p

CH(P) Spanning Paths For each finite point set in general position there exists a –open spanning path. 5 ¼ = 4 2) q 1 q 2 edge of CH bad angle ¸ 3¼ = 4 good angle · 3¼ = 4 c b q1q1 q2q2 ?

K:=CH(P\{q 1,q 2 }) Spanning Paths For each finite point set in general position there exists a –open spanning path. 5 ¼ = 4 2) q 1 q 2 edge of CH bad angle ¸ 3¼ = 4 good angle · 3¼ = 4 z b q1q1 q2q2 ? a) vertex of K in slab of q 1 q 2. E p c y

Summary o spanning tree of maxdegree three that is -open; o spanning path that is -open. Every finite planar point set in general position admits a … o triangulation that is -open; o spanning tree that is -open; 5 ¼ = 32 ¼ = 33 ¼ = 25 ¼ = 4 3 ¼ = 2 ?

Pseudotriangles Polygon with exactly 3 convex vertices (interior angle < π ).

Pseudotriangulations For a set S of n points: Partition of conv(S) into pseudo-triangles whose vertex set is exactly S.

Pseudotriangulations Minimum pseudotriangulation: n-2 pseudo-triangles Minimum  each vertex has an incident angle > π.

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