Skeda:BMonSphere.jpg

Nuk ka përmasa më të madhaja.

BMonSphere.jpg ‎((përmasa 365 × 356 px, madhësia skedës: 10 KB, lloji MIME: image/jpeg))

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PërshkrimiBMonSphere.jpg	Brownian Motion on a Sphere. The generator of ths process is ½ times the Laplace-Beltrami-Operator
Data	Summer 2007 date QS:P,+2007-00-00T00:00:00Z/9,P4241,Q40720564 (blender file as of 28.06.2007)
Burimi	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.
Autori	Thomas Steiner
Leja (Ripërdor këtë skedë)	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.

Kjo skedë është dhënë për përdorim sipas licensës Creative Commons Attribution-Share Alike 2.5 Generic.

Je i lirë të:

ta shpërndani – ta kopjoni, rishpërndani dhe përcillni punën
t’i bëni “remix” – të përshtatni punën

Sipas kushteve të mëposhtme:

atribuim – Duhet t’i jepni meritat e duhura, të siguroni një lidhje për tek licenca dhe të tregoni nëse janë bërë ndryshime. Këtë mund ta bëni në ndonjë mënyrë të arsyeshme, por jo në ndonjë mënyrë që sugjeron se licencuesi ju del zot juve apo përdorimit tuaj.
share alike – Nëse bëni një “remix”, e shndërroni, ose ndërtoni duke u bazuar te materiali, duhet t’i shpërndani kontributet tuaja sipas të njëjtës licencë ose një të tille të përputhshme me origjinalen.

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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	Data/Koha	Miniaturë	Përmasat	Përdoruesi	Koment
e tanishme	22 dhjetor 2013 21:53		365 × 356 (10 KB)	Olli Niemitalo	Cropped (in a JPEG-lossless way)
	29 shtator 2007 00:53		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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Skeda:BMonSphere.jpg

code

GNU R

blender3d

Captions

Items portrayed in this file

përshkruan

statusi i të drejtës së autorit

me të drejta autori

liçensa

Creative Commons Attribution-ShareAlike 2.5 Generic ^anglisht

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image/jpeg

checksum ^anglisht

f51c8d9194ca77a5c2a7d77d21c292e592d5c6c5

data size ^anglisht

10.693 Bajti

lartësia

356 pixel

gjerësia

365 pixel

Historiku skedës

Lidhje skedash

Përdorimi global i skedës

code

GNU R

blender3d

Captions

Items portrayed in this file

përshkruan

statusi i të drejtës së autorit

me të drejta autori

liçensa

Creative Commons Attribution-ShareAlike 2.5 Generic anglisht

lloji i medies

image/jpeg

checksum anglisht

f51c8d9194ca77a5c2a7d77d21c292e592d5c6c5

data size anglisht

10.693 Bajti

lartësia

356 pixel

gjerësia

365 pixel

Historiku skedës

Lidhje skedash

Përdorimi global i skedës

Creative Commons Attribution-ShareAlike 2.5 Generic ^anglisht

checksum ^anglisht

data size ^anglisht