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Computational Geometry• Design and analysis of
algorithms for solving geometric problems– Modern study
started with Michael Shamos in 1975
• Facility location:– geometric data
structures used to “simplify” solution procedures
-4 -3 -2 -1 0 1 2 3 4-4
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Convex Hull• Find the points that
enclose all points– Most important data
structure– Calculated, via
Graham’s scan in
-4 -3 -2 -1 0 1 2 3 4-4
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( log ), pointsO n n n
Delaunay Triangulation• Find the triangula-
tion of points that maximizes the minimum angle of any triangle– Captures proximity
relationships– Used in 3-D
animation– Calculated, via
divide and conquer, in
-4 -3 -2 -1 0 1 2 3 4-4
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( log ), pointsO n n n
Voronoi Diagram• Each region defines
area closest to a point– Open face regions
indicate points in convex hull
-4 -3 -2 -1 0 1 2 3 4-4
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-10 -5 0 5 10-10
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Voronoi Diagram• Voronoi diagram from smooshing paint between glass
– https://youtu.be/yDMtGT0b_kg
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Delaunay-Voronoi• Delaunay triangula-
tion is straight-line dual of Voronoidiagram– Can easily convert
from one to another
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Minimum Spanning Tree• Find the minimum
weight set of arcs that connect all nodes in a graph– Undirected arcs:
calculated, via Kruskal’s algorithm,
– Directed arcs:calculated, via Edmond’s branching algorithm, in
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( log ), arcs, nodesO m n m n
( ), arcs, nodesO mn m n
Kruskal’s Algorithm for MST• Algorithm:
1. Create set F of single node trees
2. Create set S of all arcs3. While S nonempty and F is
not yet spanning4. Remove min arc from S5. If removed arc connects two
different trees, then add to F, combining two trees into single tree
6. If graph connected, F forms single MST; otherwise, forms multi-tree min spanning forest
• Optimal “greedy” algorithm, runs in O(m logn)
• If directed arcs, O(mn)– useful in VRP to min vehicles– harder to code
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-4 -3 -2 -1 0 1 2 3 4-3
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1814
8
3
m = 15 arcs, n = 8 nodes
Min Spanning vs Steiner Trees • Steiner point added to reduce distance connecting three
existing points compared to min spanning tree
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b a
b
b
a a
SteinerPoint
1 3 3 , 30 60 90 triangle2 2
Min spanning tree distance > Steiner tree distance
2 3
2 3 3
2 3
4 3
b a b a
b a
a a
= ⇒ = − −
>
>
>
>
Steiner Network
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Metric Distances
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x
y
Chebychev Distances
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Proof
Great Circle Distances
13.35 mi
R
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Metric Distances using dists
× ×× ×
××
• • = = = • • • •
3 2 2 2
3 2
4 51 'mi' 'km'
dists( , , ),2 1 2 Inf3 n d m d
n m
p pD X1 X2
[ ]• • = • • = • • ⇒ • •
, 2X1 X
[ ] [ ]= • • = • • ⇒, 2X1 X
• • • • = • • = • • ⇒ • • • •
, 2X1 X
[ ]= •d
[ ]= • • •d
• • • = • • • • • •
D
• • = = ⇒ • • , 2X1 X
• • = • • D
d = 2 d = 1
[ ]• = = • • ⇒ • , 2X1 X Error
2
1
3
4
5
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Mercator Projection
-200 -150 -100 -50 0 50 100 150 200-80
-60
-40
-20
0
20
40
60
80
100
-150 -100 -50 0 50 100 150
-75
-50
-25
0
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( )
( )
proj
1proj
1proj
sinh tan
tan sinh
x x
y y
y y
−
−
=
=
=
deg radrad deg
180and180x xx xπ
π⋅
= =
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Circuity Factor
66-80 -79.5 -79 -78.5 -78 -77.5
35
35.2
35.4
35.6
35.8
36
36.2
36.4
36.6
From High Point to Goldsboro: Road = 143 mi, Great Circle = 121 mi, Circuity = 1.19
High Point
Goldsboro
road
road 1 2 1 2
: , where usually 1.15 1.5, weight of sample
( , ), estimated road distance from to
i
i
i iGC
GC
dCircuity Factor g v g v id
d g d P P P P
= ≤ ≤
≈ ⋅
∑
Estimating Circuity Factors• Circuity factor depends on both the trip density and
directness of travel network– Circuity of high trip density areas should be given more
weight when estimating overall factor for a region– Obstacles (water, mountains) limit direct road travel
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60k pop. 30k pop.
10k pop.
1,3
1,31.2
0.22
gv
==
1,2
1,2
1.1
0.67
g
v
=
=2,32,3 1.40.11
g
v
==
1 2
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