A nalysis O f Voronoi Diagrams U sing T he Geometry of salt mountains

1. Analysis Of Voronoi Diagrams Using The Geometryof salt mountains Ritsumeikanhigh school Mimura Tomohiro Miyazaki Kosuke Murata Kodai

2. 1 What is geometry of salt mountain • 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 “Geometryof salt mountain”.

3. 1 What is geometry of salt mountain

4. 2 What is voronoi diagram 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.

5. 3 the mountain ridges formed by pouring salt on various polygons

6. 3-1Triangle Same distance incenter

7. 3-2Quadrilaterals and Pentagons

8. 3-2 Examination of Quadrilaterals △ＡＢＥの傍心点 △ＡＢＥの内心点

9. 3-2 Examination of Pentagons

10. 3-3 Concave Quadrilaterals and Pentagons

11. 3-3Examination • 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.

12. 3-4 a circle board with a hole

13. 3-4 Examination ED＝EA CE＋BE ＝CE＋EA＋AB ＝CE＋ED＋AB ＝CD＋AB ＝（big circle’s radius）＋（ small circle’s radius ） ＝Constant

14. 3-5 Quadratic Curves

15. 3-5 Examination p＞PQ p＜PQ

16. 3-5 Examination To solve d which is make up (0,p) on y-axis and Q on y=x2 d If p ＜1/2, the minimum If p ＞１/2, Thus the mountain ridges are disappeared atp<1/2.

17. 3-6 One Hole

18. 3-6 Two Holes

19. 4 applications to Voronoi Diagrams

20. 4-1 Flowcharting

21. 4-2 Simulation of the program Compare to salt mountain

22. 4-2 Simulation of the program Compare to salt mountain

23. 4-3 Additively weightedVoronoi Diagrams • Weighted Voronoi Diagrams are an extension of Voronoi Diagrams. • d(x,p(i))=d(p(i))-w(i)

24. 4-4 Relation with weight and radius • salt mountains could reproduce this by replacing weight with the radius of the hole . this mean weight = radius

25. 4-5Simulation of the program Compare to salt mountain

26. 4-5Simulation of the program Compare to salt mountain

27. 5application

28. 5-1The problem of separating school districts If there are four schools in some area, like this figure, each student wants to enter the nearest of the four schools.

29. 5-3The crystal structure of molecules

30. 6conclusion • 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.

31. 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.

32. ■references • 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 MathematicsMasashi Sanaehttp://izumi-math.jp/sanae/MathTopic/gosin/gosin.htm • Function Graphing Software GRAPES KatuhisaTomodahttp://www.osaka-kyoiku.ac.jp/~tomodak/grapes/

33. Special thanks • RitsumeikanHigh School Mr,SanameMsashi • RitumeikanUniversityCollege of Science and EngineeringDr,NakajimaHisao

34. Thank you for listening !