Convex hull algorithms in 3D

Slides by: Roger Hernando

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 Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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).

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Complexity of the 3D Convex Hull

Euler’s theorem:V − E + F = 2

Triangle mesh 3F = 2E :

V − E + 2E3 = 2⇒ E = 3V − 6

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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].

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Gift wrapping

Direct extension of the 2Dalgorithm.Algorithm complexityO(nF ).

F = #convexhullfaces.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Gift wrapping

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Gift wrapping

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Gift wrapping

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Gift wrapping

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Gift wrapping

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Gift wrapping

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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).

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Merge in detail

1 Obtain a common tangent(AB) which can be computed justas in the two-dimensional case.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Merge in detail

3 The process stops when we reach again the edge AB.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Merge

The curves bounding the two convex hulls do not have to besimple.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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)}

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Merge II

until e 6= LdiscardHiddenFaces(C)

end function

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Incremental algorithm

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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 .

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Find visible region

Naive:Test every face for each point O(n2).

Conflict graph:Maintain information to aid in determining visible facets froma point.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Initialize Conflict graph

Initialize the conflict graph G with CH(P4) in linear time.Loop through all points to determine the conflicts.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Update Conflict graph

1 Discard visible facets from p by removing neighbors of p ∈ G .2 Remove p from G .3 Determine new conflicts.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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)

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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.

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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)

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Incremental algorithm

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Incremental algorithm

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Incremental algorithm

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Incremental algorithm

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

Incremental algorithm

Slides by: Roger Hernando Convex hull algorithms in 3D

Introduction Complexity Gift wrapping Divide and conquer Incremental algorithm References

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

Slides by: Roger Hernando Convex hull algorithms in 3D