1 / 34

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

A nalysis O f Voronoi Diagrams U sing T he Geometry of salt mountains. Ritsumeikan high school Mimura Tomohiro Miyazaki Kosuke Murata Kodai. 1 What is geometry of salt mountain. Mr,Kuroda suggest “the geometry of salt”

weylin
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 !

More Related