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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)