1、Analysis Of Voronoi Diagrams Using The Geometry of salt mountains,Ritsumeikan high school Mimura Tomohiro Miyazaki Kosuke Murata Kodai,When salt is sprinkled on the board which is cut into a particular shape, it creates mountain ridges The lines created by the mountain ridges can express various cha
2、racteristic Application of modeling biology by Salt Mountain Geometrics We found that Voronoi diagrams can be reproduced by salt mountains., Abstract ,2,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.
3、 We named “ Geometry of salt mountain”.,1 What is geometry of salt mountain,3,1 What is geometry of salt mountain,4,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,5,3 t
4、he mountain ridges formed by pouring salt on various polygons,6,Same distance,incenter,3-1 Triangle,7,3-2 Quadrilaterals and Pentagons,8,3-2 Examination of Quadrilaterals,内心点,傍心点,9,3-2 Examination of Pentagons,10,3-2 四角形 五角形 結果,内心円書四角形頂点一,11,3-3 Concave Quadrilaterals and Pentagons,12,The reason of
5、appearing curve line is that there are different shortest line from a concave point,Point E is same distance to line and A,There were curve lines.,3-3 Examination,13,3-3Examination,14,3-4 a circle board with a hole,15,3-4 Examination,16,EDEA CEBE CEEAAB CEEDAB CDAB （big circles radius） （ small circl
6、es radius ） Constant,3-5 Quadratic Curves,17,pPQ,pPQ,3-5 Examination,18,d,Thus the mountain ridges are disappeared at p1/2.,If p /2 ,If p 1/2 , the minimum,To solve d which is make up (0,p) on y-axis and Q on y=x2,3-5 Examination,19,3-6 One Hole,20,3-6 Two Holes,21,4 applications to Voronoi Diagrams
7、,22,4-1 Flowcharting,23,4-2 Simulation of the program Compare to salt mountain,24,4-2 Simulation of the program Compare to salt mountain,25,Weighted Voronoi Diagrams are an extension of Voronoi Diagrams.d(x, p(i) = d(p(i) - w(i),4-3 Additively weighted Voronoi Diagrams,26,salt mountains could reprod
8、uce this by replacing weight with the radius of the hole . this mean weight = radius,4-4 Relation with weight and radius,27,4-4 Relation with weight and radius,28,4-5 Simulation of the program Compare to salt mountain,29,4-5 Simulation of the program Compare to salt mountain,30,5 application,31,If t
9、here 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,32,5-2 Cell,epidermal cell of a plant （take in Ritsumeikan high school）,33,5-3 The crystal structure of molecules,34,Mountain ridges appea
10、r 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,35,7 Future
11、 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.,36, Multiplicat
12、ively weighted Voronoi Diagrams ,37,塩教幾何学 Toshiro Kuroda 折紙学幾何 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,38,Ritsumeikan High School Mr,Saname Msashi Ritumeikan University College of Science and Engineering Dr,Nakajima Hisao,Special thanks,39,Thank you for listening !,40,
