Image:BMonSphere.jpg

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This image was created with Blender.
Description

Brownian Motion on a Sphere. The generator of ths process is ½ times the Laplace-Beltrami-Operator
Source

read some papers ;) use the GNU R code and the python code (in blender3d) to create this image.
Date

summer 2007 (blender file as of 28.06.2007)
Author

Thomas Steiner
Permission
(Reusing this image)

Thomas Steiner put it under the CC-by-SA 2.5. If you use the python code or the R code, please give a reference to Christian Bayer and Thomas Steiner.


Creative Commons License
Creative Commons Attribution iconCreative Commons Share Alike icon		 This file is licensed under the Creative Commons Attribution ShareAlike 2.5 License. In short: you are free to share and make derivative works of the file under the conditions that you appropriately attribute it, and that you distribute it only under a license identical to this one. Official license

[edit] code

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

[edit] 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)


[edit] 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)

File history

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Date/TimeDimensionsUserComment
current22:53, 28 September 2007783×588 (14 KB)Thire ({{Information |Description = Brownian Motion on a Sphere |Source = read some papere ;) use the GNU R code and the python code (in blender3d) to create this image. |Date = summer 2007 (blender file as of ) |Author = Thomas Steiner |P)
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