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

-3

-2

-1

0

1

2

3

4

51

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

-3

-2

-1

0

1

2

3

4

52

( 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

-3

-2

-1

0

1

2

3

4

53

( 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

-3

-2

-1

0

1

2

3

4

-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10

54

Voronoi Diagram• Voronoi diagram from smooshing paint between glass

– https://youtu.be/yDMtGT0b_kg

55

Delaunay-Voronoi• Delaunay triangula-

tion is straight-line dual of Voronoidiagram– Can easily convert

from one to another

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

56

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

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

57

( 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

58

-4 -3 -2 -1 0 1 2 3 4-3

-2

-1

0

1

2

3

1

2

3

4

5

6

7

8

4

15

12

5

13

7

10

16

2

17

11

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

59

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

60

Metric Distances

61

x

y

Chebychev Distances

62

Proof

Great Circle Distances

13.35 mi

R

63

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

64

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

25

50

75

( )

( )

proj

1proj

1proj

sinh tan

tan sinh

x x

y y

y y

−

−

=

=

=

deg radrad deg

180and180x xx xπ

π⋅

= =

65

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

67

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