Datei:BMonSphere.jpg
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BschryybigBMonSphere.jpg | Brownian Motion on a Sphere. The generator of ths process is ½ times the Laplace-Beltrami-Operator |
Datum |
Sommer 2007 date QS:P,+2007-00-00T00:00:00Z/9,P4241,Q40720564 (blender file as of 28.06.2007) |
Quälle | 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. |
Urheber | Thomas Steiner |
Gnähmigung (Wyternutzig vu däre Datei) |
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. |
Diese Grafik wurde mit Blender erstellt.
Die Datei isch lizänziert unter dr Creative-Commons-Lizenz Namensnännig – Wytergab unter glyche Bedingige 2.5 generisch.
- Du derfsch:
- des Wärch an anderi wytergee – des Wärch kopiere, verbreite un ibertrage
- des Wärch verändere – des Wärch aapasse
- Unter däne Bedingige:
- Namenännig – Du muesch aagmässeni Aagabe zum Urheber un dr Rächt mache, e Link zue dr Lizänz zuefiege un aagee, eb Änderige gmacht wore sin. Die Aagabe chenne in ere aagmässene Art gmacht wäre, aber nit eso, ass dr Yydruck entstoht, ass dr Lizänzgeber Dii oder Dyy Gebruch vum Wärch unterstitze un guetheiße.
- Wytergab unter glyche Bedingige – Wänn du des Wärch veränderesch, umwandlesch oder druf ufböusch, no derfsch s Ergebnis nume unter dr glyche oder ere kumpatible Lizänz wie s Orginal vereffetlige.
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)
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aktuell | 21:53, 22. Dez. 2013 | 365 × 356 (10 KB) | Olli Niemitalo | Cropped (in a JPEG-lossless way) | |
00:53, 29. Sep. 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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