Analysis Of Voronoi DiagramsUsing The Geometry of salt mountains

Ritsumeikan　 high schoolMimura TomohiroMiyazaki Kosuke

Murata Kodai

Mr,Kuroda suggest　“ the geometry of salt”

When a lot of salt is poured on a board which is cut

into a particular shape, it creates a “salt mountain.

We named “ Geometry of salt mountain”. 　

1 　What is geometry of salt mountain

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1 What is geometry of salt mountain

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When some points are put like this on a diagram, a Voronoi Diagram is the diagram which separates the areas closest to each point from the other points.

2 What is voronoi diagram

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3 The mountain ridges formed by pouring salt on various polygons　

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Same distance

incenter

3-1 Triangle　

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3-2 Quadrilaterals and Pentagons　

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3-2 Examination of Quadrilaterals

△ ＡＢＥの内心点△ ＡＢＥの傍心点

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3-2 Examination of Pentagons

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3-3 Concave Quadrilaterals and Pentagons

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The reason of appearing curve line is that there are different shortest line from a concave point

Point E is same distance to

line ｌ and A

There were curve lines.

3-3 Examination

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A

E

F

line l

3-4 a circle board with a hole

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3-4 Examination

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ED ＝ EACE ＋ BE＝ CE ＋ EA ＋ AB＝ CE ＋ ED ＋ AB＝ CD ＋ AB＝（ big circle’s radius ） ＋（ small circle’s radius ）＝ Constant

3-5 Quadratic Curves

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p ＜ PQp ＞ PQ

3-5 Examination

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d

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221

)21()(2

2

2242222

ppx

pxpxpxxd

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pd

Thus the mountain ridges are disappeared at p<1/2.

If p ＞１ /2 ,

If p ＜ 1/2 , the minimum

pd

To solve d which is make up (0,p) on y-axis and Q on y=x2

3-5 Examination

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21

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ppp

3-6 One Hole

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3-6 Two Holes

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4 APPLICATIONS TO 　　　　　　VORONOI DIAGRAMS

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4-1 Flowcharting

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Start

Set the range and domain (x0,y0)- (x1,y1)Set the number of point num

Set the coordinates of point (AX,AY)Set the radius which is r of circle

Set the color ct

Loop1From y0 to y1 about y

Loop2From x0 to x1 about x

Loop3From i=0 to num

L(i)=SQR((X-AX(i))^2+(Y-AY(i))^2)-r(i)

i=0NO

YES

MIN=L(i)ct=i

L(i)<MINNO

YES

MIN=L(i)ct=i

Give color which is ct 2 to point

SET POINT STYLE 1 PLOT POINTS: x,y

Loop3

Loop2

Loop1

Loop4From i=0 to num

Radius ｒ(i) middle(AX(i),AY(i))

such circle was drown

Loop4

End

4-2 Simulation of the program Compare to salt mountain

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4-2 Simulation of the program Compare to salt mountain

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Weighted Voronoi Diagrams are an extension of Voronoi Diagrams.

d(x, p(i)) = d(p(i)) - w(i)

4-3 Additively weighted Voronoi Diagrams

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salt mountains could reproduce this by replacing weight with the radius of the hole . this mean weight = radius

4-4 Relation with weight and radius

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4-5 Simulation of the program Compare to salt mountain

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4-5 Simulation of the program Compare to salt mountain

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5　 Application

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If there are four schools in some area, like this figure, each student wants to enter the nearest of the four schools.

5-1 The problem of separating school districts

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5-3 The crystal structure of molecules 　

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Mountain ridges appear where the distances to the nearest side is shared by two or more sides.

The prediction of the program matches the mountain ridge lines and the additively weighted Voronoi Diagram also matches the program.

Salt mountain can reproduce various phenomenon in biology and physics.

6　 Conclusion

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7 Future plan

We want to analyze mountain ridge lines in various

shapes.

We could reproduce additively weighted Voronoi

Diagrams so we research how to reproduce

Multiplicatively weighted Voronoi Diagrams.

We want to be able to create the shape of the board to

match any given mountain ridges.

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塩が教える幾何学　　　 Toshiro Kuroda

折り紙で学ぶなわばりの幾何　 Konichi Kato

Spring of Mathematics 　 Masashi Sanae

http://izumi-math.jp/sanae/MathTopic/gosin/gosin.htm

Function Graphing Software GRAPES Katuhisa

Tomoda

http://www.osaka-kyoiku.ac.jp/~tomodak/grapes/

■ References

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塩が教える幾何学

折り紙で学ぶなわばりの幾何

Ritsumeikan High School

Mr,Saname Msashi

Ritumeikan University 　College of Science and Engineering

　　　　 Dr,Nakajima Hisao

SPECIAL THANKS

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Thank you for listening !

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