Датотека:BMonSphere.jpg
BMonSphere.jpg (365 × 356 пиксела, величина датотеке: 10 kB, MIME тип: image/jpeg)
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ОписBMonSphere.jpg | Brownian Motion on a Sphere. The generator of ths process is ½ times the Laplace-Beltrami-Operator |
Датум |
Summer 2007 date QS:P,+2007-00-00T00:00:00Z/9,P4241,Q40720564 (blender file as of 28.06.2007) |
Извор | 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. |
Аутор | Thomas Steiner |
Дозвола (Поновно коришћење ове датотеке) |
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. |
This image was created with Blender.
Ова датотека је доступна под лиценцом Creative Commons Ауторство-Делити под истим условима 2.5 Генеричка.
- Дозвољено је:
- да делите – да умножавате, расподељујете и преносите дело
- да прерађујете – да прерадите дело
- Под следећим условима:
- ауторство – Морате да дате одговарајуће заслуге, обезбедите везу ка лиценци и назначите да ли су измене направљене. Можете то урадити на било који разуман манир, али не на начин који предлаже да лиценцатор одобрава вас или ваше коришћење.
- делити под истим условима – Ако измените, преобразите или доградите овај материјал, морате поделити своје доприносе под истом или компатибилном лиценцом као оригинал.
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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Датум/време | Минијатура | Димензије | Корисник | Коментар | |
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тренутна | 21:53, 22. децембар 2013. | 365 × 356 (10 kB) | Olli Niemitalo | Cropped (in a JPEG-lossless way) | |
00:53, 29. септембар 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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- Употреба на kn.wikipedia.org
- Употреба на lt.wikipedia.org
- Употреба на ms.wikipedia.org
- Употреба на pt.wikipedia.org
- Употреба на ru.wikipedia.org
- Употреба на sl.wikipedia.org
- Употреба на sq.wikipedia.org
- Употреба на uk.wikipedia.org
- Употреба на www.wikidata.org
Преузето из „https://sr.wikipedia.org/wiki/Датотека:BMonSphere.jpg”