By Dor Lahav. Overview Straight Skeletons Convex Polygons Constrained Voronoi diagrams and Delauney...
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- By Dor Lahav
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- Overview Straight Skeletons Convex Polygons Constrained Voronoi
diagrams and Delauney triangulations
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- Straight Skeletons
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- Straight Skeleton- Motivation The occurrence of curved edges in
the line segment Voronoi diagram V(G) is a disadvantage in the
computer representation. a different alternative to V(G)- the
Straight Skeleton.
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- Straight Skeleton
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- A few definitions: - A connected component of G is called a
figure. - Every simple polygon which rises from a figure, is called
a wavefront.
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- Straight Skeleton There are two types of changes: - Edge event:
a wavefront edge collapses to length zero. - Split event: a
wavefront edge splits due to interference or self interference. -
After each type of event, we have a new set of wavefronts.
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- Straight Skeleton The edges of S(G) are pieces of angular
bisectors traced out by wavefront vertices. Each vertex of S(G)
corresponds to event. So we get a unique structure defining a
polygonal partition of a plane.
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- Straight Skeleton Lets see an example
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- Straight Skeleton How many faces in the diagram? Each segment
of G gives rise to two wavefront edges and thus to two faces, one
on each side of the segment. Each terminal of G gives rise to one
face. This gives a total of 2m+t = O(n) faces(m-edges, t-
terminals). There is also an exact bound on the number of
vertices
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- Straight Skeleton Lemma: Let G be a planar straight graph on n
points, t of which are terminals. The number of(finite and
infinite) vertices of S(G) is exactly 2n+t-2.
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- Straight Skeleton The Straight Skeleton has a 3- dimensional
interpretation obtained by defining the height of a point x in the
plane as the unique time when x is reached by the wavefront. Thats
how S(G) lifts out a polygonal surface. Points of G have height
zero.
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- Straight Skeleton
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- Convex Polygons
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- Let C be a convex n-gon in the plane. The medial axis M(C) of C
is a tree whose edges are pieces of angular bisectors of Cs sides.
There is a simple randomized incremental algorithm that computes
M(C) in O(n) time.
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- Convex Polygons
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- Done!
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- Running time? M(C) has an upper bound of 2n-3 edges. Each edge
belongs to two faces. Hence, the average number of edges of a face
is Thats a constant time per face. So, a total running time of
O(n).
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- Convex Polygons- Voronoi diagram
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- Constrained Voronoi diagrams and Delauney triangulations
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- Constrained Voronoi diagrams Let S be a set of n point sites in
the plane. Let L be a set of non-crossing line segments spanned by
S. We define the bounded distance between two points x and y in the
plane as:
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- Constrained Voronoi diagrams For each segment l, the regions
clipped by l from the right are extended to the left of l, as if
only the sites of these regions were present. The regions clipped
by l from the left are extended similarly.
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- Constrained Delauney triangulations If we dualize now by
connecting sites of regions that share an edge, a full
triangulation that includes L is obtained.
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