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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
4
1
2
21
)21()(2
2
2242222
pp
x
pxpxpxxd
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1 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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2
1
4
1 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)<MIN
NO
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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