この画像（あるいは、このカテゴリーやこのページにある全ての画像）は、JPEG形式でアップロードされました。 しかしながら、PNG形式、あるいはSVG形式で保存すれば、もっと効率的に、もっと正確に情報を含めることができます。もし可能であれば、JPEG圧縮を経ていないデータを元に、または現在のファイルにあるノイズを除去して、この画像を圧縮ノイズのないPNG形式またはSVG形式で再アップロードしてください。

 

この 画像はBlenderで作成されました。

このファイルはクリエイティブ・コモンズ 表示-継承 2.5 一般ライセンスのもとに利用を許諾されています。

あなたは以下の条件に従う場合に限り、自由に

• 共有 – 本作品を複製、頒布、展示、実演できます。
• 再構成 – 二次的著作物を作成できます。

あなたの従うべき条件は以下の通りです。

• 表示 – あなたは適切なクレジットを表示し、ライセンスへのリンクを提供し、変更があったらその旨を示さなければなりません。これらは合理的であればどのような方法で行っても構いませんが、許諾者があなたやあなたの利用行為を支持していると示唆するような方法は除きます。
• 継承 – もしあなたがこの作品をリミックスしたり、改変したり、加工した場合には、あなたはあなたの貢献部分を元の作品とこれと同一または互換性があるライセンスの下に頒布しなければなりません。

https://creativecommons.org/licenses/by-sa/2.5CC BY-SA 2.5 Creative Commons Attribution-Share Alike 2.5 truetrue

code

Perhaps you grab the source from the "edit" page without the wikiformating.

GNU R

This creates the paths and saves them into textfiles that can be read by blender. There are also paths for BMs on a torus.

# calculate a Brownian motion on the sphere; the output is a list # consisting of: # Z ... BM on the sphere # Y ... tangential BM, see Price&Williams # b ... independent 1D BM (see Price & Williams) # B ... generating 3D BM # n ... number of time-steps in the discretization # T ... the above processes are given on a uniform mesh of size #       n on [0,T] euler = function(x0, T, n) {   # initialize objects   dt = T/(n-1);   dB = matrix(rep(0,3*(n-1)),ncol=3, nrow=n-1);   dB[,1] = rnorm(n-1, 0, sqrt(dt));   dB[,2] = rnorm(n-1, 0, sqrt(dt));   dB[,3] = rnorm(n-1, 0, sqrt(dt));   Z = matrix(rep(0,3*n), ncol=3, nrow=n);   dZ = matrix(rep(0,3*(n-1)), ncol=3, nrow=n-1);   Y = matrix(rep(0,3*n), ncol=3, nrow=n);   B = matrix(rep(0,3*n), ncol=3, nrow=n);   b = rep(0, n);   Z[1,] = x0;    #do the computation   for(k in 2:n){     B[k,] = B[k-1,] + dB[k-1,];     dZ[k-1,] = cross(Z[k-1,],dB[k-1,]) - Z[k-1,]*dt;     Z[k,] = Z[k-1,] + dZ[k-1,];     Y[k,] = Y[k-1,] - cross(Z[k-1,],dZ[k-1,]);     b[k] = b[k-1] + dot(Z[k-1,],dB[k-1,]);   }   return(list(Z = Z, Y = Y, b = b, B = B, n = n, T = T)); }  # write the output from euler in csv-files euler.write = function(bms, files=c("Z.csv","Y.csv","b.csv","B.csv"),steps=bms$n){   bigsteps=round(seq(1,bms$n,length=steps))   write.table(bms$Z[bigsteps,],file=files[1],col.names=F,row.names=F,sep=",",dec=".");   write.table(bms$Y[bigsteps,],file=files[2],col.names=F,row.names=F,sep=",",dec=".");   write.table(bms$b[bigsteps],file=files[3],col.names=F,row.names=F,sep=",",dec=".");   write.table(bms$B[bigsteps,],file=files[4],col.names=F,row.names=F,sep=",",dec="."); }  # calculate a Brownian motion on a 3-d torus with outer # radius R and inner radius r eulerTorus = function(x0, r, R, t, n) {   # initialize objects   dt = t/(n-1);   dB = matrix(rep(0,3*(n-1)),ncol=3, nrow=n-1);   dB[,1] = rnorm(n-1, 0, sqrt(dt));   dB[,2] = rnorm(n-1, 0, sqrt(dt));   dB[,3] = rnorm(n-1, 0, sqrt(dt));   Z = matrix(rep(0,3*n), ncol=3, nrow=n);   B = matrix(rep(0,3*n), ncol=3, nrow=n);   dZ = matrix(rep(0,3*(n-1)), ncol=3, nrow=n-1);   Z[1,] = x0;   nT = rep(0,3);    #do the computation   for(k in 2:n){     B[k,] = B[k-1,] + dB[k-1,];     nT = nTorus(Z[k-1,],r,R);     dZ[k-1,] = cross(nT, dB[k-1,]) + HTorus(Z[k-1,],r,R)*nT*dt;     Z[k,] = Z[k-1,] + dZ[k-1,];   }   return(list(Z = Z, B = B, n = n, t = t)); }  # write the output from euler in csv-files torus.write = function(bmt, files=c("tZ.csv","tB.csv"),steps=bmt$n){   bigsteps=round(seq(1,bmt$n,length=steps))   write.table(bmt$Z[bigsteps,],file=files[1],col.names=F,row.names=F,sep=",",dec=".");   write.table(bmt$B[bigsteps,],file=files[2],col.names=F,row.names=F,sep=",",dec="."); }  # "defining" function of a torus fTorus = function(x,r,R){   return((x[1]^2+x[2]^2+x[3]^2+R^2-r^2)^2 - 4*R^2*(x[1]^2+x[2]^2)); }  # normal vector of a 3-d torus with outer radius R and inner radius r nTorus = function(x, r, R) {   c1 = x[1]*(x[1]^2+x[2]^2+x[3]^2-R^2-r^2)/(3*x[1]^4*x[2]^2+3*x[3]^4*x[2]^2     +3*x[3]^4*x[1]^2+6*x[3]^2*x[1]^2*x[2]^2+3*x[1]^2*x[2]^4+3*x[3]^2*x[2]^4     -2*x[3]^2*R^2*r^2-4*x[1]^2*x[2]^2*R^2+x[1]^6+x[2]^6+x[3]^6+3*x[3]^2*x[1]^4     -4*x[1]^2*x[2]^2*r^2-4*x[1]^2*x[3]^2*r^2+2*R^2*x[1]^2*r^2     -4*x[2]^2*x[3]^2*r^2+2*R^2*x[2]^2*r^2-2*x[1]^4*R^2-2*x[1]^4*r^2     +R^4*x[1]^2+x[1]^2*r^4-2*x[2]^4*R^2-2*x[2]^4*r^2+R^4*x[2]^2+x[2]^2*r^4     +x[3]^2*R^4+x[3]^2*r^4-2*x[3]^4*r^2+2*x[3]^4*R^2)^(1/2);   c2 = x[2]*(x[1]^2+x[2]^2+x[3]^2-R^2-r^2)/(3*x[1]^4*x[2]^2+3*x[3]^4*x[2]^2     +3*x[3]^4*x[1]^2+6*x[3]^2*x[1]^2*x[2]^2+3*x[1]^2*x[2]^4+3*x[3]^2*x[2]^4     -2*x[3]^2*R^2*r^2-4*x[1]^2*x[2]^2*R^2+x[1]^6+x[2]^6+x[3]^6     +3*x[3]^2*x[1]^4-4*x[1]^2*x[2]^2*r^2-4*x[1]^2*x[3]^2*r^2+2*R^2*x[1]^2*r^2     -4*x[2]^2*x[3]^2*r^2+2*R^2*x[2]^2*r^2-2*x[1]^4*R^2-2*x[1]^4*r^2+R^4*x[1]^2     +x[1]^2*r^4-2*x[2]^4*R^2-2*x[2]^4*r^2+R^4*x[2]^2+x[2]^2*r^4+x[3]^2*R^4     +x[3]^2*r^4-2*x[3]^4*r^2+2*x[3]^4*R^2)^(1/2);   c3 = (x[1]^2+x[2]^2+x[3]^2+R^2-r^2)*x[3]/(3*x[1]^4*x[2]^2+3*x[3]^4*x[2]^2                                             +3*x[3]^4*x[1]^2                                             +6*x[3]^2*x[1]^2*x[2]^2                                             +3*x[1]^2*x[2]^4+3*x[3]^2*x[2]^4                                             -2*x[3]^2*R^2*r^2                                             -4*x[1]^2*x[2]^2*R^2+x[1]^6                                             +x[2]^6+x[3]^6+3*x[3]^2*x[1]^4                                             -4*x[1]^2*x[2]^2*r^2                                             -4*x[1]^2*x[3]^2*r^2                                             +2*R^2*x[1]^2*r^2                                             -4*x[2]^2*x[3]^2*r^2                                             +2*R^2*x[2]^2*r^2-2*x[1]^4*R^2                                             -2*x[1]^4*r^2+R^4*x[1]^2                                             +x[1]^2*r^4-2*x[2]^4*R^2                                             -2*x[2]^4*r^2+R^4*x[2]^2                                             +x[2]^2*r^4+x[3]^2*R^4                                             +x[3]^2*r^4-2*x[3]^4*r^2                                             +2*x[3]^4*R^2)^(1/2);   return(c(c1,c2,c3)); }  # mean curvature of a 3-d torus with outer radius R and inner radius r HTorus = function(x, r, R){   return( -(3*x[1]^4*r^4+4*x[2]^6*x[3]^2+4*x[1]^6*x[2]^2-3*x[2]^4*x[3]^2*R^2             -2*x[1]^6*R^2+4*x[1]^2*x[3]^6+x[3]^6*R^2+4*x[2]^4*R^2*r^2-x[1]^2*r^6             -x[2]^2*r^6+x[2]^4*R^4+4*x[2]^2*x[3]^2*R^4+6*x[2]^2*x[3]^2*r^4             -2*x[1]^2*R^2*r^4-x[1]^2*R^4*r^2-9*x[1]^4*x[2]^2*r^2             -9*x[1]^4*x[3]^2*r^2+4*x[1]^4*R^2*r^2+12*x[1]^2*x[3]^4*x[2]^2             -3*x[2]^6*r^2+4*x[1]^6*x[3]^2+3*x[3]^4*r^4-x[3]^4*R^4             -9*x[2]^4*x[3]^2*r^2+2*x[2]^2*x[3]^2*R^2*r^2+4*x[1]^2*x[2]^6             -6*x[1]^2*x[3]^2*x[2]^2*R^2-x[3]^2*r^6+6*x[2]^4*x[3]^4+x[3]^8             +x[1]^8+x[2]^8-3*x[1]^6*r^2+6*x[1]^4*x[3]^4+12*x[1]^2*x[3]^2*x[2]^4             -6*x[1]^2*x[2]^4*R^2-2*x[3]^4*R^2*r^2-2*x[2]^2*R^2*r^4-x[2]^2*R^4*r^2             -9*x[2]^2*x[3]^4*r^2+x[3]^2*R^2*r^4+x[3]^2*R^4*r^2-9*x[1]^2*x[2]^4*r^2             +2*x[1]^2*R^4*x[2]^2+6*x[1]^2*x[2]^2*r^4-3*x[1]^4*x[3]^2*R^2             -6*x[1]^4*x[2]^2*R^2+4*x[1]^2*x[3]^2*R^4+6*x[1]^2*x[3]^2*r^4             -9*x[1]^2*x[3]^4*r^2+8*x[1]^2*R^2*x[2]^2*r^2+2*x[1]^2*x[3]^2*R^2*r^2             +x[1]^4*R^4-3*x[3]^6*r^2-2*x[2]^6*R^2+6*x[1]^4*x[2]^4-x[3]^2*R^6             -18*x[1]^2*x[2]^2*x[3]^2*r^2+4*x[2]^2*x[3]^6+12*x[1]^4*x[3]^2*x[2]^2             +3*x[2]^4*r^4)/(3*x[1]^4*x[2]^2+3*x[3]^4*x[2]^2+3*x[3]^4*x[1]^2                             +6*x[3]^2*x[1]^2*x[2]^2+3*x[1]^2*x[2]^4+3*x[3]^2*x[2]^4                             -2*x[3]^2*R^2*r^2-4*x[1]^2*x[2]^2*R^2+x[1]^6+x[2]^6                             +x[3]^6+3*x[3]^2*x[1]^4-4*x[1]^2*x[2]^2*r^2                             -4*x[1]^2*x[3]^2*r^2+2*R^2*x[1]^2*r^2                             -4*x[2]^2*x[3]^2*r^2+2*R^2*x[2]^2*r^2-2*x[1]^4*R^2                             -2*x[1]^4*r^2+R^4*x[1]^2+x[1]^2*r^4-2*x[2]^4*R^2                             -2*x[2]^4*r^2+R^4*x[2]^2+x[2]^2*r^4+x[3]^2*R^4                             +x[3]^2*r^4-2*x[3]^4*r^2+2*x[3]^4*R^2)^(3/2)); }  # calculate the cross product of the two 3-dim vectors # x and y. No argument-checking for performance reasons cross = function(x,y){   res = rep(0,3);   res[1] = x[2]*y[3] - x[3]*y[2];   res[2] = -x[1]*y[3] + x[3]*y[1];   res[3] = x[1]*y[2] - x[2]*y[1];   return(res); }  # calculate the inner product of two vectors of dim 3 # returns a number, not a 1x1-matrix! dot = function(x,y){   return(sum(x*y)); }  # calculate the cross product of the two 3-dim vectors # x and y. No argument-checking for performance reasons cross = function(x,y){   res = rep(0,3);   res[1] = x[2]*y[3] - x[3]*y[2];   res[2] = -x[1]*y[3] + x[3]*y[1];   res[3] = x[1]*y[2] - x[2]*y[1];   return(res); }  ############# ### main-teil set.seed(280180) et=eulerTorus(c(3,0,0),3,5,19,10000) torus.write(et,steps=9000) # #bms=euler(c(1,0,0),4,70000) #euler.write(bms,steps=10000)  

blender3d

The blender (python) code to create a image that looks almost like this one. Play around...

## import data from matlab-text-file and draw BM on the S^2  ## (c) 2007 by Christan Bayer and Thomas Steiner  from Blender import Curve, Object, Scene, Window, BezTriple, Mesh, Material, Camera, World from math import *  ##import der BM auf der Kugel aus einem csv-file def importcurve(inpath="Z.csv"):         infile = open(inpath,'r')         lines = infile.readlines()         vec=[]         for i in lines:                 li=i.split(',')                 vec.append([float(li[0]),float(li[1]),float(li[2].strip())])         infile.close()         return(vec)  ##function um aus einem vektor (mit den x,y,z Koordinaten) eine Kurve zu machen def vec2Cur(curPts,name="BMonSphere"):         bztr=[]         bztr.append(BezTriple.New(curPts[0]))         bztr[0].handleTypes=(BezTriple.HandleTypes.VECT,BezTriple.HandleTypes.VECT)         cur=Curve.New(name) ##TODO wenn es das Objekt schon gibt, dann nicht neu erzeugen         cur.appendNurb(bztr[0])         for i in range(1,len(curPts)):                 bztr.append(BezTriple.New(curPts[i]))                 bztr[i].handleTypes=(BezTriple.HandleTypes.VECT,BezTriple.HandleTypes.VECT)                 cur[0].append(bztr[i])         return( cur )  #erzeugt einen kreis, der später die BM umgibt (liegt in y-z-Ebene) def circle(r,name="tubus"):         bzcir=[]         bzcir.append(BezTriple.New(0.,-r,-4./3.*r, 0.,-r,0., 0.,-r,4./3.*r))         bzcir[0].handleTypes=(BezTriple.HandleTypes.FREE,BezTriple.HandleTypes.FREE)         cur=Curve.New(name) ##TODO wenn es das Objekt schon gibt, dann nicht neu erzeugen         cur.appendNurb(bzcir[0])         #jetzt alle weietren pkte         bzcir.append(BezTriple.New(0.,r,4./3.*r, 0.,r,0., 0.,r,-4./3.*r))         bzcir[1].handleTypes=(BezTriple.HandleTypes.FREE,BezTriple.HandleTypes.FREE)         cur[0].append(bzcir[1])         bzcir.append(BezTriple.New(0.,-r,-4./3.*r, 0.,-r,0., 0.,-r,4./3.*r))         bzcir[2].handleTypes=(BezTriple.HandleTypes.FREE,BezTriple.HandleTypes.FREE)         cur[0].append(bzcir[2])         return ( cur )  #erzeuge mit skript eine (glas)kugel (UVSphere) def sphGlass(r=1.0,name="Glaskugel",n=40,smooth=0):         glass=Mesh.New(name) ##TODO wenn es das Objekt schon gibt, dann nicht neu erzeugen         for i in range(0,n):                 for j in range(0,n):                         x=sin(j*pi*2.0/(n-1))*cos(-pi/2.0+i*pi/(n-1))*1.0*r                         y=cos(j*pi*2.0/(n-1))*(cos(-pi/2.0+i*pi/(n-1)))*1.0*r                         z=sin(-pi/2.0+i*pi/(n-1))*1.0*r                         glass.verts.extend(x,y,z)         for i in range(0,n-1):                  for j in range(0,n-1):                         glass.faces.extend([i*n+j,i*n+j+1,(i+1)*n+j+1,(i+1)*n+j])                         glass.faces[i*(n-1)+j].smooth=1         return( glass )  def torus(r=0.3,R=1.4):          krGro=circle(r=R,name="grTorusKreis")           #jetzt das material ändern def verglasen(mesh):         matGlass = Material.New("glas") ##TODO wenn es das Objekt schon gibt, dann nicht neu erzeugen         #matGlass.setSpecShader(0.6)         matGlass.setHardness(30) #für spec: 30         matGlass.setRayMirr(0.15)         matGlass.setFresnelMirr(4.9)         matGlass.setFresnelMirrFac(1.8)         matGlass.setIOR(1.52)         matGlass.setFresnelTrans(3.9)         matGlass.setSpecTransp(2.7)         #glass.materials.setSpecTransp(1.0)         matGlass.rgbCol = [0.66, 0.81, 0.85]         matGlass.mode |= Material.Modes.ZTRANSP         matGlass.mode |= Material.Modes.RAYTRANSP         #matGlass.mode |= Material.Modes.RAYMIRROR         mesh.materials=[matGlass]         return ( mesh )  def maleBM(mesh):         matDraht = Material.New("roterDraht") ##TODO wenn es das Objekt schon gibt, dann nicht neu erzeugen         matDraht.rgbCol = [1.0, 0.1, 0.1]         mesh.materials=[matDraht]         return( mesh )  #eine solide Mesh-Ebene (Quader) # auf der höhe ebh, dicke d, seitenlänge (quadratisch) 2*gr def ebene(ebh=-2.5,d=0.1,gr=6.0,name="Schattenebene"):         quader=Mesh.New(name) ##TODO wenn es das Objekt schon gibt, dann nicht neu erzeugen         #obere ebene         quader.verts.extend(gr,gr,ebh)         quader.verts.extend(-gr,gr,ebh)         quader.verts.extend(-gr,-gr,ebh)         quader.verts.extend(gr,-gr,ebh)         #untere ebene         quader.verts.extend(gr,gr,ebh-d)         quader.verts.extend(-gr,gr,ebh-d)         quader.verts.extend(-gr,-gr,ebh-d)         quader.verts.extend(gr,-gr,ebh-d)         quader.faces.extend([0,1,2,3])         quader.faces.extend([0,4,5,1])         quader.faces.extend([1,5,6,2])         quader.faces.extend([2,6,7,3])         quader.faces.extend([3,7,4,0])         quader.faces.extend([4,7,6,5])         #die ebene einfärben         matEb = Material.New("ebenen_material") ##TODO wenn es das Objekt schon gibt, dann nicht neu erzeugen         matEb.rgbCol = [0.53, 0.51, 0.31]         matEb.mode |= Material.Modes.TRANSPSHADOW         matEb.mode |= Material.Modes.ZTRANSP         quader.materials=[matEb]         return (quader)  ################### #### main-teil ####  # wechsel in den edit-mode editmode = Window.EditMode() if editmode: Window.EditMode(0)  dataBMS=importcurve("C:/Dokumente und Einstellungen/thire/Desktop/bmsphere/Z.csv") #dataBMS=importcurve("H:\MyDocs\sphere\Z.csv") BMScur=vec2Cur(dataBMS,"BMname") #dataStereo=importcurve("H:\MyDocs\sphere\stZ.csv") #stereoCur=vec2Cur(dataStereo,"SterName")  cir=circle(r=0.01)  glass=sphGlass() glass=verglasen(glass) ebe=ebene()  #jetzt alles hinzufügen scn=Scene.GetCurrent() obBMScur=scn.objects.new(BMScur,"BMonSphere") obcir=scn.objects.new(cir,"round") obgla=scn.objects.new(glass,"Glaskugel") obebe=scn.objects.new(ebe,"Ebene") #obStereo=scn.objects.new(stereoCur,"StereoCurObj")  BMScur.setBevOb(obcir) BMScur.update() BMScur=maleBM(BMScur)  #stereoCur.setBevOb(obcir) #stereoCur.update()  cam = Object.Get("Camera")  #cam.setLocation(-5., 5.5, 2.9)  #cam.setEuler(62.0,-1.,222.6) #alternativ, besser?? cam.setLocation(-3.3, 8.4, 1.7)  cam.setEuler(74,0,200)  world=World.GetCurrent() world.setZen([0.81,0.82,0.61]) world.setHor([0.77,0.85,0.66])  if editmode: Window.EditMode(1)  # optional, zurück n den letzten modus           #ergebnis von #set.seed(24112000) #sbm=euler(c(0,0,-1),T=1.5,n=5000) #euler.write(sbm)