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Covex hull algorithms in 3D
Slides by: Roger Hernando
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Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References
Outline
1 Introduction
2 Complexity
3 Gift wrapping
4 Divide and conquer
5 Incremental algorithm
6 References
Slides by: Roger Hernando Covex hull algorithms in 3D
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Problem statement
Given P: set of n points in 3D.Convex hull of P: CH(P), thesmallest polyhedron s.t. allelements of P on or in the interiorof CH(P).
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Visibility test
A point is visible from a face?The point is above or below the supporting plane of the face.The volume of the tetrahedron is positive or negative.
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Complexity of the 3D Convex Hull
Euler’s theorem:V − E + F = 2
Triangle mesh 3F = 2E :
V − E + 2E3 = 2⇒ E = 3V − 6
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Complexity of the Convex Hull
Given a set S of n points in Rn what is maximum #edges onCH(S)?
Bernard Chazelle [1990]: CH of n points in Rd in optimalworst-case is O
(n log n + nb d
2 c)
. Bound by Seidel[1981].
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Gift wrapping
Direct extension of the 2Dalgorithm.Algorithm complexityO(nF ).
F = #convexhullfaces.
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Gift wrapping
Ensure: H = Convex hull of point-set PRequire: point-set P
H = ∅(p1, p2) = HullEdge(P)Q = {(p1, p2)}while Q 6= ∅ do
(p1, p2) = Q.pop()if notProcessed((p1, p2)) then
p3 = triangleVertexWithAllPointsToTheLeft(p1, p2)H = H ∪ {(p1, p2, p3)}Q = Q ∪ {(p2, p3), (p3, p1)}markProcessedEdge((p1, p2))
end ifend while
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Gift wrapping
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Gift wrapping
Slides by: Roger Hernando Covex hull algorithms in 3D
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Gift wrapping
Slides by: Roger Hernando Covex hull algorithms in 3D
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Gift wrapping
Slides by: Roger Hernando Covex hull algorithms in 3D
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Gift wrapping
Slides by: Roger Hernando Covex hull algorithms in 3D
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Gift wrapping
Slides by: Roger Hernando Covex hull algorithms in 3D
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Divide and conquer
The same idea of the 2d algorithm.1 Split the point-set P in two sets.2 Recursively split, construct CH, and merge.
Merge takes O(n)⇒ Algorithm complexity O(n log n).
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Divide and conquer
Ensure: Convex hull of point-set PRequire: point-set P(sorted by certain coord)
function divideAndConquer(P)if |P| ≤ x then
return incremental(P)end if(P1,P2) = split(P)H1 = divideAndConquer(P1)H2 = divideAndConquer(P2)return merge(H1,H2)
end function
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Merge
Determine a supporting line of the convex hulls, projecting thehulls and using the 2D algorithm.Use wrapping algorithm to create the additional faces in orderto construct a cylinder of triangles connecting the hulls.Remove the hidden faces hidden by the wrapped band.
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Merge in detail
1 Obtain a common tangent(AB) which can be computed justas in the two-dimensional case.
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Merge in detail
2 We next need to find a triangle ABC whose vertex C mustbelong to either the left hull or the right hull. Consequently,either AC is an edge of the left hull or BC is an edge of theright hull.
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Merge in detail
3 Now we have a new edge AC that joins the left hull and theright hull. We can find triangle ACD just as we did ABC.
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Merge in detail
3 The process stops when we reach again the edge AB.
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Merge in detail
Merge can be performed in linear time O(n), because the circularorder that follow vertex on intersection of new convex hull edges,with a separating plane H0.
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Merge
The curves bounding the two convex hulls do not have to besimple.
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Merge I
Ensure: C is the convex hull of H1, H2Require: Convex hulls H1, H2
function merge(H1, H2)C = H1 ∪ H2e(p1, p2) = supportingLine(H1, H2)L = erepeat
p3 = giftWrapAroundEdge(e)if p3 ∈ H1 then
e = (p2, p3)else
e = (p1, p3)end ifC = C ∪ {(p1, p2, p3)}
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Merge II
until e 6= LdiscardHiddenFaces(C)
end function
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Incremental algorithm
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Incremental algorithm
1 Create tetrahedron(initial CH).pick 2 points p1 and p2.pick a point not lying on the line p1p2.pick a point not lying on the plane p1p2p3(if not found use a2D algorithm).
2 Randomize the remaining points P.3 For each pi ∈ P, add pi into the CHi−1
if pi lies inside or on the boundary of CHi−1 then do nothing.if pi lies outside of CHi−1 insert pi .
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Incremental algorithm
Ensure: C Convex hull of point-set PRequire: point-set P
C = findInitialTetrahedron(P)P = P − Cfor all p ∈ P do
if p outside C thenF = visbleFaces(C , p)C = C − FC = connectBoundaryToPoint(C , p)
end ifend for
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Find visible region
Naive:Test every face for each point O(n2).
Conflict graph:Maintain information to aid in determining visible facets froma point.
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Conflict graph
Relates unprocessed points with facets of CH they can see.
Bipartite graph.pt unprocessed points.f faces of the convex hull.conflict arcs, point pi sees fj .
Maintain sets:Pconflict(f ), points that can see f .Fconflict(p), faces visible from p(visibleregion deleted after insertion of p).
At any step of the algorithm, we know allconflicts between the remaining points and
facets on the current CH.
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Initialize Conflict graph
Initialize the conflict graph G with CH(P4) in linear time.Loop through all points to determine the conflicts.
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Update Conflict graph
1 Discard visible facets from p by removing neighbors of p ∈ G .2 Remove p from G .3 Determine new conflicts.
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Determine new conflicts
Check if Pconflict(f2) is in conflict with f .Check if Pconflict(f1) is in conflict with f .If pr is coplanar with f1, f has the same conflicts as f1.
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Incremental algorithm I
Ensure: C Convex hull of point-set PRequire: point-set P
C = findInitialTetrahedron(P)initializeConflictGraph(P,C)P = P − Cfor all p ∈ P do
if Fconflict(p) 6= ∅ thendelete(Fconflict(p), C)L = borderEdges(C)for all e ∈ L do
f = createNewTriangularFace(e, p)f ′ = neighbourFace(e)addFaceToG(f )if areCoplanar(f , f ′) then
Pconflict(f ) = Pconflict(f ′)else
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Incremental algorithm II
P(e) = Pcoflict(f1) ∪ Pcoflict(f2)for all p ∈ P(e) do
if isVisble(f , p) thenaddConflict(p, f )
end ifend for
end ifend for
end ifend for
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Analysis
We want to bound the total number of facets created/deleted bythe algorithm.
#faces4 +
∑nr=5 E |deg(pr ,CH(Pr ))| ≤ 4 + 6(n − 4) = 6n − 20 = O(n)
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Analysis
We want to bound the total number of Points which can seehorizon edges at any stage of the algorithm(
∑e card(P(e))).
Define:Set of active configurations τ(S).A flap ∆ = (p, q, s, t).∆ ∈ τ(S)all edges in the flap are edges of the CH(S).Set K (∆) points which have the flap ∆ as horizon edge.
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Analysis
We want to bound the total number of Points which can seehorizon edges at any stage of the algorithm(
∑e card(P(e))).∑
e card(P(e)) ≤∑
∆ card(K (∆)) ≤∑n
r=1 16(n−r
r) (6r−12
r
)≤
≤ 96n ln n = O(n log n)
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Incremental algorithm
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Incremental algorithm
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Incremental algorithm
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Incremental algorithm
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Incremental algorithm
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References I
Applet:http://www.cse.unsw.edu.au/ lambert/java/3d/hull.htmM. de Berg, O. Cheong, M. van Kreveld, M. Overmars,Computational Geometry Algorithms and ApplicationsJ. O’Rourke,Computational Geometry in CPartha P. Goswami, Introduction to Computational GeometryDavid M. Mount, Lecture notes.Bernard Chazelle, An optimal convex hull algorith in any fixeddimensionAlexander Wolf, SlidesJason C. Yang, SlidesPeter Felkel, Slides
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