File:BMonSphere.jpg
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DescriptionBMonSphere.jpg | Brownian Motion on a Sphere. The generator of ths process is ½ times the Laplace-Beltrami-Operator |
Date |
Summer 2007 date QS:P,+2007-00-00T00:00:00Z/9,P4241,Q40720564 (blender file as of 28.06.2007) |
Source | read some papers (eg Price, Gareth C.; Williams, David: "Rolling with “slipping”" : I. Séminaire de probabilités de Strasbourg, 17 (1983), p. 194-197 You can download it from http://www.numdam.org/item?id=SPS_1983__17__194_0) use the GNU R code and the python code (in blender3d) to create this image. |
Author | Thomas Steiner |
Permission (Reusing this file) |
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. |
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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)
Captions
Brownian Motion on a Sphere, as a process generated by the Laplace-Beltrami-Operator
Броуновское движение на сфере Генератор этого процесса в ½ раза превышает оператор Лапласа-Бельтрами.
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10,693 byte
356 pixel
365 pixel
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 19:53, 22 December 2013 | ![]() | 365 × 356 (10 KB) | Olli Niemitalo | Cropped (in a JPEG-lossless way) |
22:53, 28 September 2007 | ![]() | 783 × 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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