UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2004

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UMass Lowell Computer Science 91.504 Advanced
AlgorithmsComputational Geometry Prof. Karen
Daniels Spring, 2004

• Chapter 4 3D Convex Hulls
• Monday, 2/23/04

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Chapter 4 3D Convex Hulls

• Polyhedra
• Algorithms
• Implementation
• Polyhedral Boundary Representations
• Randomized Incremental Algorithm
• Higher Dimensions

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Polyhedra What are they?

• Polyhedron generalizes 2D polygon to 3D
• Consists of flat polygonal faces
• Boundary/surface contains
• 0D vertices 1D edges 2D faces
• Components intersect properly. Each face pair
• disjoint or have vertex in common or have
  vertex/edge/vertex in common
• Local topology is proper
• at every point neighborhood is homeomorphic to
  disk
• Global topology is proper
• connected, closed, bounded
• may have holes

polyhedra
not polyhedra!
for more examples, see http//www.ScienceU.com/geo
metry/facts/solids/handson.html
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Polyhedra A Flawed Definition

• Definition of Polyhedron1
• A polyhedron1 is a region of space bounded by a
  finite set of polygons such that
• every polygon shares at least one edge with some
  other polygon
• every edge is shared by exactly two polygons.

polyhedra and polyhedra1?
not polyhedra and not polyhedra1?
HW Problem - What is wrong with this
definition? - Find an example of an object that
is a polyhedron1 but is not a polyhedron.
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Polyhedra A Resource
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Polyhedra Regular Polytopes

• Convex polyhedra are polytopes
• Regular polyhedra are polytopes that have
• regular faces, faces, solid (dihedral) angles
• There are exactly 5 regular polytopes

Excellent math references by H.S.M. Coxeter -
Introduction to Geometry (2nd edition), Wiley
Sons, 1969 - Regular Polytopes, Dover
Publications, 1973
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Polyhedra Eulers Formula
V - E F 2

• Proof has 3 parts
• 1) Convert polyhedron surface to planar graph
• 2) Tree theorem
• 3) Proof by induction

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Algorithms 2D Gift Wrapping

• Use one extreme edge as an anchor for finding the
  next

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Algorithms 3D Gift Wrapping
O(n2) time output sensitive O(nF) for F faces
on hull
CxHull Animations http//www.cse.unsw.edu.au/lam
bert/java/3d/hull.html
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Algorithms 2D Divide-and-Conquer

• Divide-and-Conquer in a geometric setting
• O(n) merge step is the challenge
• Find upper and lower tangents
• Lower tangent find rightmost pt of A leftmost
  pt of B then walk it downwards
• Idea is extended to 3D in Chapter 4.

B
A
Algorithm 2D DIVIDE-and-CONQUER Sort points by x
coordinate Divide points into 2 sets A and B A
contains left n/2 points B contains right n/2
points Compute ConvexHull(A) and ConvexHull(B)
recursively Merge ConvexHull(A) and ConvexHull(B)
O(nlgn)
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Algorithms 3D Divide and Conquer
O(n log n) time !
CxHull Animations http//www.cse.unsw.edu.au/lam
bert/java/3d/hull.html
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Algorithms 2D QuickHull

• Concentrate on points close to hull boundary
• Named for similarity to Quicksort

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Algorithms 3D QuickHull
CxHull Animations http//www.cse.unsw.edu.au/lam
bert/java/3d/hull.html
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Algorithms Qhull (gt 2D )
http//www.geom.umn.edu/software/qhull/
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Algorithms 2D Incremental

• Add points, one at a time
• update hull for each new point
• Key step becomes adding a single point to an
  existing hull.
• Idea is extended to 3D in Chapter 4.

Algorithm 2D INCREMENTAL ALGORITHM Let H2
ConvexHullp0 , p1 , p2 for k 3 to n - 1
do Hk ConvexHull Hk-1 U pk
O(n2)
can be improved to O(nlgn)
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Algorithms 3D Incremental

• Add points, one at a time
• Update hull Q for each new point p
• Case 1 p is in existing hull Q
• Discard p
• Case 2 p is not in existing hull Q
• Compute cone tangent to Q whose apex is p
• Cone consists of triangular tangent faces
• Base of each is an edge of Q
• Construct new hull using cone

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Algorithms 3D Incremental
Algorithm 3D INCREMENTAL ALGORITHM Let H3
ConvexHull p0 , p1 , p2, p3
tetrahedron for i 4 to n - 1 do for each
face f of Hi-1 do compute volume of tetrahedron
determined by f and pi mark f visible iff
volume lt 0 if no faces are visible then
Discard pi (it is inside Hi-1) else
for each border edge e of Hi-1 do Construct
cone face determined by e and pi for
each visible face f do Delete f
Update Hi
O(n2)
Randomized incremental has expected O(nlgn) time.
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Algorithms 3D Incremental
O(n2) time
CxHull Animations http//www.cse.unsw.edu.au/lam
bert/java/3d/hull.html
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3D Visibility

• To build our 3D visibility intuition
• T planar triangle in 3D
• C closed region bounded by a 3D cube
• frequent problem in graphics

C
T
HW Problem What is the largest number of
vertices P can have for any triangle?
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Polyhedral Boundary Representations
v7
v2
v1
e1-
v1
es twin
e1
f1
f1
v3
e
v6
e
f0
e0,1
f0
e0
e4,0
v0
v5
v0
e0-
v4
twin edge DCEL doubly connected edge list
winged edge

• represent edge as 2 halves
• lists vertex, face, edge/twin
• more storage space
• facilitates face traversal
• can represent holes with face inner/outer edge
  pointer

• focus is on edge
• edge orientation is arbitrary

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Polyhedral Boundary Representations Quad-Edge
e1,1
e0,0

• general subdivision of oriented 2D manifold
• edge record is part of
• endpoint 1 list
• endpoint 2 list
• face A list
• face B list