塩山幾何学を用いたボロノイ図の解析

Ritsumeikan high school
Mimura Tomohiro
Miyazaki 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”.
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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.
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Same
distance
incenter
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△ＡＢＥの傍心点
△ＡＢＥの内心点
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
The reason of appearing curve line is that there are
different shortest line from a concave point
A
E
line l
F
Point E is same distance to
line ｌ and A
There were curve lines.
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ED＝EA
CE＋BE
＝CE＋EA＋AB
＝CE＋ED＋AB
＝CD＋AB
＝（big circle’s radius）
＋（ small circle’s radius ）
＝Constant
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p＞PQ
p＜PQ
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To solve d which is make up (0,p) on yaxis and Q on y=x2
d 2  x 2  ( x 2  p ) 2  x 4  (1  2 p ) x 2  p 2
1 2 p 
1

  x2 
  p
2 
4

2
p
p
d
1

4
p
1
2
If p ＜1/2 , the minimum
If p ＞１/2 ,
d
dp
p
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Thus the mountain ridges are disappeared
at p<1/2.
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Start
i=0
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
NO
Loop3
YES
Loop2
MIN=L(i)
ct=i
Loop1
From y0 to y1 about y
Loop1
L(i)<MIN
Loop2
From x0 to x1 about x
Loop3
From i=0 to num
L(i)=SQR((X-AX(i))^2
+(Y-AY(i))^2)-r(i)
NO
Loop4
From i=0 to num
YES
MIN=L(i)
ct=i
Give color which is ct 2 to
point
SET POINT STYLE 1
PLOT POINTS: x,y
Radius ｒ(i) middle
(AX(i),AY(i))
such circle was drown
Loop4
End
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
Weighted Voronoi Diagrams are an extension of Voronoi
Diagrams.

d(x, p(i)) = d(p(i)) - w(i)
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
salt mountains could reproduce this by replacing
weight with the radius of the hole . this mean
weight = radius
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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.
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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.
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
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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Toshiro Kuroda

塩が教える幾何学
塩が教える幾何学

折り紙で学ぶなわばりの幾何
Konichi Kato
折り紙で学ぶなわばりの幾何

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/
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
Ritsumeikan High School
Mr,Saname Msashi

Ritumeikan University
College of Science and Engineering
Dr,Nakajima Hisao
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