B ISECTOR G RAPHS FOR M IN -/M AX -V OLUME R OOFS OVER S IMPLE P OLYGONS Günther Eder – Martin Held – Peter Palfrader March 2016, Lugano
M OTIVATION r Comparing two polygons. A lower area does not always lead to a lower roof volume. r The lower envelope over all planes is not the minimum volume roof. (Neither does the upper envelope lead to the maximum volume roof.)
M OTIVATION r Comparing two polygons. A lower area does not always lead to a lower roof volume. r The lower envelope over all planes is not the minimum volume roof. (Neither does the upper envelope lead to the maximum volume roof.) x/y x/z
I NTRODUCTION A PPROACH r Building on Roof Model and Bisector Graphs [2] . r Gradient Property [2] generalized. r Wavefront Propagation [1] extended by two additional events. 2. Oswin Aichholzer, Franz Aurenhammer, David Alberts, and Bernd Gärtner. A Novel Type of Skeleton for Polygons. Journal of Universal Computer Science , 1995 1. Oswin Aichholzer and Franz Aurenhammer. Straight Skeletons for General Polygonal Figures in the Plane. In Proc. 2nd Internat. Comput. and Combinat. Conf. Springer Berlin Heidelberg, 1996
I NTRODUCTION A PPROACH r Building on Roof Model and Bisector Graphs [2] . r Gradient Property [2] generalized. r Wavefront Propagation [1] extended by two additional events. T HEOREM (R OOF ↔ BISECTOR GRAPH [2] ) Every roof for P corresponds to a unique bisector graph of P, and vice versa. 2. Oswin Aichholzer, Franz Aurenhammer, David Alberts, and Bernd Gärtner. A Novel Type of Skeleton for Polygons. Journal of Universal Computer Science , 1995 1. Oswin Aichholzer and Franz Aurenhammer. Straight Skeletons for General Polygonal Figures in the Plane. In Proc. 2nd Internat. Comput. and Combinat. Conf. Springer Berlin Heidelberg, 1996
I NTRODUCTION A PPROACH r Building on Roof Model and Bisector Graphs [2] . r Gradient Property [2] generalized. r Wavefront Propagation [1] extended by two additional events. N ATURAL G RADIENT P ROPERTY Let R ( P ) be a roof for P . We say that a facet f of R ( P ) has the natural gradient property if, for every point p ∈ f , there exists a path that (i) starts at p , (ii) follows the steepest gradient, and (iii) reaches the boundary of P . 2. Oswin Aichholzer, Franz Aurenhammer, David Alberts, and Bernd Gärtner. A Novel Type of Skeleton for Polygons. Journal of Universal Computer Science , 1995 1. Oswin Aichholzer and Franz Aurenhammer. Straight Skeletons for General Polygonal Figures in the Plane. In Proc. 2nd Internat. Comput. and Combinat. Conf. Springer Berlin Heidelberg, 1996
I NTRODUCTION A PPROACH r Building on Roof Model and Bisector Graphs [2] . r Gradient Property [2] generalized. r Wavefront Propagation [1] extended by two additional events. E XTENDED W AVEFRONT P ROPAGATION r Edge Event and Split Event [2] . r Create Event and Divide Event 2. Oswin Aichholzer, Franz Aurenhammer, David Alberts, and Bernd Gärtner. A Novel Type of Skeleton for Polygons. Journal of Universal Computer Science , 1995 1. Oswin Aichholzer and Franz Aurenhammer. Straight Skeletons for General Polygonal Figures in the Plane. In Proc. 2nd Internat. Comput. and Combinat. Conf. Springer Berlin Heidelberg, 1996
I NTRODUCTION A PPROACH r Building on Roof Model and Bisector Graphs [2] . r Gradient Property [2] generalized. r Wavefront Propagation [1] extended by two additional events. E XTENDED W AVEFRONT P ROPAGATION r Edge Event and Split Event [2] . r Create Event and Divide Event 2. Oswin Aichholzer, Franz Aurenhammer, David Alberts, and Bernd Gärtner. A Novel Type of Skeleton for Polygons. Journal of Universal Computer Science , 1995 1. Oswin Aichholzer and Franz Aurenhammer. Straight Skeletons for General Polygonal Figures in the Plane. In Proc. 2nd Internat. Comput. and Combinat. Conf. Springer Berlin Heidelberg, 1996
I NTRODUCTION A PPROACH r Building on Roof Model and Bisector Graphs [2] . r Gradient Property [2] generalized. r Wavefront Propagation [1] extended by two additional events. G ENERAL P OSITION r No two edges of P are parallel to each other. r Not more than three bisectors of edges of P meet in one point. e e ′ 2. Oswin Aichholzer, Franz Aurenhammer, David Alberts, and Bernd Gärtner. A Novel Type of Skeleton for Polygons. Journal of Universal Computer Science , 1995 1. Oswin Aichholzer and Franz Aurenhammer. Straight Skeletons for General Polygonal Figures in the Plane. In Proc. 2nd Internat. Comput. and Combinat. Conf. Springer Berlin Heidelberg, 1996
I NTRODUCTION A PPROACH r Building on Roof Model and Bisector Graphs [2] . r Gradient Property [2] generalized. r Wavefront Propagation [1] extended by two additional events. G ENERAL P OSITION r No two edges of P are parallel to each other. r Not more than three bisectors of edges of P meet in one point. e e ′ 2. Oswin Aichholzer, Franz Aurenhammer, David Alberts, and Bernd Gärtner. A Novel Type of Skeleton for Polygons. Journal of Universal Computer Science , 1995 1. Oswin Aichholzer and Franz Aurenhammer. Straight Skeletons for General Polygonal Figures in the Plane. In Proc. 2nd Internat. Comput. and Combinat. Conf. Springer Berlin Heidelberg, 1996
I NTRODUCTION A PPROACH r Building on Roof Model and Bisector Graphs [2] . r Gradient Property [2] generalized. r Wavefront Propagation [1] extended by two additional events. D EFINITION (M IN -/M AX -V OLUME B ISECTOR G RAPH ) The maximum-volume bisector graph B max ( P ) of a polygon P is a bisector graph B ( P ) where the associated roof R ( P ) has the natural gradient property for each of its facets and that maximizes the volume over all possible natural roofs for P . Similarly for the minimum-volume bisector graph B min ( P ) . 2. Oswin Aichholzer, Franz Aurenhammer, David Alberts, and Bernd Gärtner. A Novel Type of Skeleton for Polygons. Journal of Universal Computer Science , 1995 1. Oswin Aichholzer and Franz Aurenhammer. Straight Skeletons for General Polygonal Figures in the Plane. In Proc. 2nd Internat. Comput. and Combinat. Conf. Springer Berlin Heidelberg, 1996
B ISECTOR Two consecutive edges e i , e j of P . e i e j
B ISECTOR Edges of P are oriented. A half plane Π( e ) that starts at the supporting line ℓ ( e ) of an edge spans to its left. Π( e ) overlaps locally with the interior of P . Π( e i ) ∩ Π( e j ) e i e j
B ISECTOR A bisector b i , j spans from the intersection of the supporting line of two edges into their common interior. b i , j e i e j
C REATE E VENT Wavefront propagation of e i and e j . b i , j e i e j
C REATE E VENT Wavefront propagation of e i and e j . A wavefront edge moves at unit speed (self parallel). The speed s ( v ) of a wavefront vertex v depends on the angle between the supporting lines forming its bisector [3] . b i , j v i , j 1 s ( v i , j ) = sin( α/ 2) α e i e j 3. Siu-Wing Cheng and Antoine Vigneron. Motorcycle Graphs and Straight Skeletons. In Proc. 13th Symposium on Discrete Algorithms , 2002
C REATE E VENT Every bisector defines a vertex that has a starting point and associated speed. In case such a vertex is not part of the wavefront we call it stealth vertex . b i , j v i , j 1 s ( v i , j ) = sin( α/ 2) 1 s ( v j , k ) = sin( β/ 2) α v j , k e i e j β e k
C REATE E VENT Another input edge e x of P . b i , j e x e i e j
C REATE E VENT At some point p i , j , x is the wavefront vertex incident with the supporting line from the wavefront edge of e x . b i , j p i , j , x e x e i e j
C REATE E VENT At some point p i , j , x is the wavefront vertex incident with the supporting line from the wavefront edge of e x . The three bisectors meet at that point as well. b i , x b j , x e x b i , j e i e j
C REATE E VENT The wavefront changes: an additional edge e is created, and e is parallel to the wavefront edge of e x . The two wavefront vertices on b i , x and b j , x are both reflex. b i , x b j , x e x b i , j e i e j
C REATE E VENT The wavefront changes: an additional edge e is created, and e is parallel to the wavefront edge of e x . The two wavefront vertices on b i , x and b j , x are both reflex. Π e x b i , x b j , x e x b i , j e i e j
C REATE E VENT , CONT . Consecutive edges along a polygon boundary. e x e j e i
C REATE E VENT , CONT . Consecutive edges along a polygon boundary. Wavefront propagation on the first (edge) event. e x e j e i
C REATE E VENT , CONT . Consecutive edges along a polygon boundary. Wavefront propagation on the first (edge) event. Wavefront propagation continues. e x e j e i
C REATE E VENT , CONT . The stealth vertex v i , j becomes incident with the wavefront edge originating from e x at point p i , j , x . e x b i , j e j e i
C REATE E VENT , CONT . The stealth vertex v i , j becomes incident with the wavefront edge originating from e x at point p i , j , x . Three arcs start at this point and create two new facets. b i , x b j , x p i , j , x e x b i , j e j e i
C REATE E VENT , CONT . The stealth vertex v i , j becomes incident with the wavefront edge originating from e x at point p i , j , x . Three arcs start at this point and create two new facets. One of these facets lies in the plane Π( e i ) and one in Π( e j ) . Π( e i ) Π( e j ) Π( e x ) e x e j e i
A CCELERATING /D ECELERATING C REATE E VENT L EMMA A small disc c centered around a create event p is partitioned into three wedges by the three arcs incident at p. If one wedge has an angle greater than π it involves a wavefront vertex, starting at p, that moves faster than the wavefront vertex which ends at p.
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