Download - Ball Packings and Fat Voronoi Diagrams

Ball Packings and

Fat Voronoi Diagrams

Don Sheehy

Ball Packings

Ball Packings

Packing unit balls into a square:

Ball Packings

Packing unit balls into a square:Packing means disjoint interiors.

Ball Packings

Packing unit balls into a square:Packing means disjoint interiors.Maximal means no more will fit.

Ball Packings


Goal: Find a small, maximal packing.

Ball Packings


Goal: Find a small, maximal packing.

Greedy Algorithm: - Find the biggest empty space. - Add another ball there.

Problem:

Given n balls in a s x s box, how many total will there be after running the greedy algorithm?

Problem:


Upper Bound

Problem:


Upper Bound

Total volume is at most s2.

Problem:


Upper Bound


s2

!

Problem:


Upper Bound


(Packing Argument)s2

!

Problem:


Upper Bound Lower Bound



!

Problem:



Total volume is at most s2. Each Voronoi cell is contained in a ball of radius 2.


!

Problem:





!

s2

4!

Problem:




(Packing Argument) (Covering Argument)s2

!

s2

4!

Problem:




(Packing Argument) (Covering Argument)

Why care about a factor of 4? Because 4 = 2d

.

s2

!

s2

4!

Voronoi Aspect Ratio


Ball Packing

r ! 1, R < 2


Ball Packing

r ! 1, R < 2R

r< constant


Ball Packing Meshing

r ! 1, R < 2R

r< constant



r ! 1, R < 2R

r< constant

Good Aspect Ratio Voronoi Diagram

Good Quality Delaunay Triangulation



r ! 1, R < 2R

r< constant

Good Aspect Ratio Voronoi Diagram

Good Quality Delaunay Triangulation

Voronoi Refinement Meshing:While any Voronoi cell has “bad” aspect ratio,add its farthest corner.

The Local Feature Size

f(x) = distance to second nearest point of P.

P: inputM: output



P: inputM: output

Key fact about Greedy Voronoi refinement:

for all vertices v in M.

rv ! f(v) ! krv



P: inputM: output

Key fact about Greedy Voronoi refinement:

for all vertices v in M.

rv ! f(v) ! krv

The local feature size tells us how big the ball for v is.

The Upper Bound

The Upper BoundTheorem: |M | < (k+1)d

!d

!B

1f(x)d dx.

The Upper BoundTheorem: |M | < (k+1)d

!d

!B

1f(x)d dx.

proof.

The Upper Bound

!B

1

f(x)ddx =

"v!M

!Vor(v)

1

f(x)ddx

>"v!M

!bv

1

f(x)ddx

!"v!M

!bv

1

((k + 1)rv)ddx

="v!M

rdv!d

((k + 1)rv)d

= |M |!d

(k + 1)d.

Theorem: |M | < (k+1)d

!d

!B

1f(x)d dx.

proof.

The Upper Bound

Recall: f(v) ! krv.

!B

1

f(x)ddx =

"v!M

!Vor(v)

1

f(x)ddx

>"v!M

!bv

1

f(x)ddx

!"v!M

!bv

1

((k + 1)rv)ddx

="v!M

rdv!d

((k + 1)rv)d

= |M |!d

(k + 1)d.


!d

!B

1f(x)d dx.

proof.

The Upper Bound

Recall: f(v) ! krv.

!B

1

f(x)ddx =

"v!M

!Vor(v)

1

f(x)ddx

>"v!M

!bv

1

f(x)ddx

!"v!M

!bv

1

((k + 1)rv)ddx

="v!M

rdv!d

((k + 1)rv)d

= |M |!d

(k + 1)d.


!d

!B

1f(x)d dx.

proof.

Claim: If x ! bv,then f(x) " (k + 1)rv.

The Lower Bound

The Lower BoundTheorem: |M | > 1

kd!d

!B

1f(x)d dx.


kd!d

!B

1f(x)d dx.

proof.

!B

1

f(x)ddx =

"v!M

!Vor(v)

1

f(x)ddx

!"v!M

!Vor(v)

1

rdvdx

<"v!M

!kbv

1

rdvdx

="v!M

kdrdv!d

rdv

= |M |kd!d.


kd!d

!B

1f(x)d dx.

proof.

!B

1

f(x)ddx =

"v!M

!Vor(v)

1

f(x)ddx

!"v!M

!Vor(v)

1

rdvdx

<"v!M

!kbv

1

rdvdx

="v!M

kdrdv!d

rdv

= |M |kd!d.

The Lower Bound

Claim: If x ! Vor(v) then f(x) " rv.

Theorem: |M | > 1kd!d

!B

1f(x)d dx.

proof.

!B

1

f(x)ddx =

"v!M

!Vor(v)

1

f(x)ddx

!"v!M

!Vor(v)

1

rdvdx

<"v!M

!kbv

1

rdvdx

="v!M

kdrdv!d

rdv

= |M |kd!d.

The Lower Bound

Claim: If x ! Vor(v) then f(x) " rv.

Theorem: |M | > 1kd!d

!B

1f(x)d dx.

Claim: Vor(v) ! kbv.

proof.

Degree BoundsHow many neighbors can any point have?


!

q!Q

vol(bq) >!

q!Q

vol

"

bp

k

#

= |Q|$rp

k

%d

!d.


!

q!Q

vol(bq) < vol(3kbp) = (3krp)d!d.

!

q!Q

vol(bq) >!

q!Q

vol

"

bp

k

#

= |Q|$rp

k

%d

!d.


!

q!Q


!

q!Q

vol(bq) >!

q!Q

vol

"

bp

k

#

= |Q|$rp

k

%d

!d.

! |Q| < 3d


!

q!Q


!

q!Q

vol(bq) >!

q!Q

vol

"

bp

k

#

= |Q|$rp

k

%d

!d.

! |Q| < 3d

Total Voronoi Edges < 3d|M|


!

q!Q


!

q!Q

vol(bq) >!

q!Q

vol

"

bp

k

#

= |Q|$rp

k

%d

!d.

! |Q| < 3d

Total Voronoi Edges < 3d|M|

Same tricks as before!(Packing Argument)

Fat Voronoi DiagramsIdea: Let the center shift.


Good Aspect Ratio Fat


Good Aspect Ratio Fat

Fat Voronoi Conjecture:

The number of neighbors of any cell in a fat Voronoi diagram is 2O(d).

Fat Voronoi diagrams in the plane.

Fat Voronoi diagrams in the plane.Here’s the trick.


!q

Here’s the trick.


!q

!q

Here’s the trick.


!q

!q

2! =

!

q!Q

"q =

!

q!Q

#q

Here’s the trick.


!q

!q

2! =

!

q!Q

"q =

!

q!Q

#q

Here’s the trick.

Is this new?


!q

!q

2! =

!

q!Q

"q =

!

q!Q

#q

Here’s the trick.

Is this new?

Punchline: Fat Voronoi Conjecture Holds in the plane.


!q

!q

2! =

!

q!Q

"q =

!

q!Q

#q

Here’s the trick.

Is this new?

Punchline: Fat Voronoi Conjecture Holds in the plane.

3D?

Thank you.

Download - Ball Packings and Fat Voronoi Diagrams

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