{"id":419,"date":"2013-05-30T17:13:16","date_gmt":"2013-05-30T16:13:16","guid":{"rendered":"http:\/\/numbercrunch.de\/blog\/?p=419"},"modified":"2023-01-18T23:37:32","modified_gmt":"2023-01-18T22:37:32","slug":"visualizing-streamlines","status":"publish","type":"post","link":"https:\/\/www.numbercrunch.de\/blog\/2013\/05\/visualizing-streamlines\/","title":{"rendered":"Visualizing streamlines"},"content":{"rendered":"

In Visualizing vector fields<\/a> I showed how to plot vector fields using Python and Matplotlib. Streamlines are a concept that is closely related to vector fields. Mathematically speaking streamlines are\u00a0continuous lines whose tangent at each point is given by a vector field. Each line, and therefore also streamlines, can be parametrized by some parameter $t$. A streamline $\\vec{r}(t)$ fulfils the equation\\begin{equation}
\n\\frac{\\mathrm{d}\\vec{r}(t)}{\\mathrm{d}t} = g(t)\\vec{E}(\\vec{r}(t))\\,,
\n\\end{equation}
\nwhere $\\vec{E}(\\vec{r}(t))$ is the vector field and $g(t)$ some scaling function. The scaling functions is arbitrary but must not be zero. It basically determines how fast one moves along the streamline as a function of the parameter $t$. It is often convenient to set\\begin{equation}
\ng(t)=\\frac{1}{|\\vec{E}(\\vec{r}(t))|}\\,.
\n\\end{equation}<\/p>\n

Since version 1.2.0 the Python package Matplotlib comes with a streamplot<\/a>\u00a0 function for quick and easy visualizing two-dimensional streamlines. Coming back the example of an electric dipole from\u00a0Visualizing vector fields<\/a> the following Python code plots the streamlines of an electric dipole. Compared to the previous post on the plotting vector fields this code is somewhat more generic. First some charges are specified and afterwards the total electric field is calculated by summing over the electric field of the individual charges.<\/p>\n

#!\/usr\/bin\/env python\n\n# import useful modules\nimport matplotlib \nfrom numpy import *\nfrom pylab import *\nfrom scipy.integrate import ode\n\n# use LaTeX, choose nice some looking fonts and tweak some settings\nmatplotlib.rc('font', family='serif')\nmatplotlib.rc('font', size=16)\nmatplotlib.rc('legend', fontsize=16)\nmatplotlib.rc('legend', numpoints=1)\nmatplotlib.rc('legend', handlelength=1.5)\nmatplotlib.rc('legend', frameon=False)\nmatplotlib.rc('xtick.major', pad=7)\nmatplotlib.rc('xtick.minor', pad=7)\nmatplotlib.rc('text', usetex=True)\nmatplotlib.rc('text.latex', \n              preamble=[r'\\usepackage[T1]{fontenc}',\n                        r'\\usepackage{amsmath}',\n                        r'\\usepackage{txfonts}',\n                        r'\\usepackage{textcomp}'])\n\nclass charge:\n    def __init__(self, q, pos):\n        self.q=q\n        self.pos=pos\n\ndef E_point_charge(q, a, x, y):\n    return q*(x-a[0])\/((x-a[0])**2+(y-a[1])**2)**(1.5), \\\n        q*(y-a[1])\/((x-a[0])**2+(y-a[1])**2)**(1.5)\n\ndef E_total(x, y, charges):\n    Ex, Ey=0, 0\n    for C in charges:\n        E=E_point_charge(C.q, C.pos, x, y)\n        Ex=Ex+E[0]\n        Ey=Ey+E[1]\n    return [ Ex, Ey ]\n\nclose('all')\nfigure(figsize=(6, 4.5))\n\n# charges and positions\ncharges=[ charge(1, [-1, 0]), charge(-1, [1, 0]) ]\n\n# plot field lines\nx0, x1=-2, 2\ny0, y1=-1.5, 1.5\nx=linspace(x0, x1, 64)\ny=linspace(y0, y1, 64)\nx, y=meshgrid(x, y)\nEx, Ey=E_total(x, y, charges)\nstreamplot(x, y, Ex, Ey, color='k')\n\n# plot point charges\nfor C in charges:\n    if C.q>0:\n        plot(C.pos[0], C.pos[1], 'bo', ms=8*sqrt(C.q))\n    if C.q<0:\n        plot(C.pos[0], C.pos[1], 'ro', ms=8*sqrt(-C.q))\n\nxlabel('$x$')\nylabel('$y$')\ngca().set_xlim(x0, x1)\ngca().set_ylim(y0, y1)\nshow()\naxis('image')\nsavefig('visualization_streamlines_1.png')<\/pre>\n
\"Streamlines<\/a>
Streamlines of an electric dipole visualized using Matplotlib’s streamplot function.<\/figcaption><\/figure>\n
Matplotlib’s streamplot<\/code> function is very generic and easy to use. However it does not know anything about specific characteristics of the vector field to plot. For example, it is not able to take into account that streamlines of electric fields always start and end at the charges. Therefore, the following code plots streamlines by solving the streamlines’ ordinary differential equations. We always start close in the vicinity of the electric charges and extend each streamline until it has reached another charge or has left the plotting area.<\/p>\n
#!\/usr\/bin\/env python\n\n# import usefull modules\nimport matplotlib \nfrom numpy import *\nfrom pylab import *\nfrom scipy.integrate import ode\n\n# use LaTeX, choose nice some looking fonts and tweak some settings\nmatplotlib.rc('font', family='serif')\nmatplotlib.rc('font', size=16)\nmatplotlib.rc('legend', fontsize=16)\nmatplotlib.rc('legend', numpoints=1)\nmatplotlib.rc('legend', handlelength=1.5)\nmatplotlib.rc('legend', frameon=False)\nmatplotlib.rc('xtick.major', pad=7)\nmatplotlib.rc('xtick.minor', pad=7)\nmatplotlib.rc('text', usetex=True)\nmatplotlib.rc('text.latex', \n              preamble=[r'\\usepackage[T1]{fontenc}',\n                        r'\\usepackage{amsmath}',\n                        r'\\usepackage{txfonts}',\n                        r'\\usepackage{textcomp}'])\n\nclass charge:\n    def __init__(self, q, pos):\n        self.q=q\n        self.pos=pos\n\ndef E_point_charge(q, a, x, y):\n    return q*(x-a[0])\/((x-a[0])**2+(y-a[1])**2)**(1.5), \\\n        q*(y-a[1])\/((x-a[0])**2+(y-a[1])**2)**(1.5)\n\ndef E_total(x, y, charges):\n    Ex, Ey=0, 0\n    for C in charges:\n        E=E_point_charge(C.q, C.pos, x, y)\n        Ex=Ex+E[0]\n        Ey=Ey+E[1]\n    return [ Ex, Ey ]\n\ndef E_dir(t, y, charges):\n    Ex, Ey=E_total(y[0], y[1], charges)\n    n=sqrt(Ex**2+Ey*Ey)\n    return [Ex\/n, Ey\/n]\n\nclose('all')\nfigure(figsize=(6, 4.5))\n\n# charges and positions\ncharges=[ charge(1, [-1, 0]), charge(-1, [1, 0]) ]\n\n# plot field lines\nx0, x1=-2, 2\ny0, y1=-1.5, 1.5\nR=0.01\n# loop over all charges\nfor C in charges:\n    # plot field lines starting in current charge\n    dt=0.8*R\n    if C.q<0:\n        dt=-dt\n    # loop over field lines starting in different directions \n    # around current charge\n    for alpha in linspace(0, 2*pi*15\/16, 16):\n        r=ode(E_dir)\n        r.set_integrator('vode')\n        r.set_f_params(charges)\n        x=[ C.pos[0] + cos(alpha)*R ]\n        y=[ C.pos[1] + sin(alpha)*R ]\n        r.set_initial_value([x[0], y[0]], 0)\n        while r.successful():\n            r.integrate(r.t+dt)\n            x.append(r.y[0])\n            y.append(r.y[1])\n            hit_charge=False\n            # check if field line left drwaing area or ends in some charge\n            for C2 in charges:\n                if sqrt((r.y[0]-C2.pos[0])**2+(r.y[1]-C2.pos[1])**2)<R:\n                    hit_charge=True\n            if hit_charge or (not (x0<r.y[0] and r.y[0]<x1)) or \\\n                    (not (y0<r.y[1] and r.y[1]<y1)):\n                break\n        plot(x, y, '-k')\n\n# plot point charges\nfor C in charges:\n    if C.q>0:\n        plot(C.pos[0], C.pos[1], 'bo', ms=8*sqrt(C.q))\n    if C.q<0:\n        plot(C.pos[0], C.pos[1], 'ro', ms=8*sqrt(-C.q))\n\nxlabel('$x$')\nylabel('$y$')\ngca().set_xlim(x0, x1)\ngca().set_ylim(y0, y1)\nshow()\naxis('image')\nsavefig('visualization_streamlines_2.png')<\/pre>\n\n\n\n\n
\"Streamlines\"<\/a><\/td>\n<\/tr>\n
Streamlines of an electric dipole visualized by solving the streamlines’ differential equations.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"
In Visualizing vector fields I showed how to plot vector fields using Python and Matplotlib. Streamlines are a concept that is closely related to vector fields. Mathematically speaking streamlines are\u00a0continuous lines whose tangent at each point is given by a vector field. Each line, and therefore also streamlines, can be parametrized by some parameter $t$.… Continue reading Visualizing streamlines<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-419","post","type-post","status-publish","format-standard","hentry","category-python"],"yoast_head":"\nVisualizing streamlines - Number Crunch<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.numbercrunch.de\/blog\/2013\/05\/visualizing-streamlines\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Visualizing streamlines - Number Crunch\" \/>\n<meta property=\"og:description\" content=\"In Visualizing vector fields I showed how to plot vector fields using Python and Matplotlib. Streamlines are a concept that is closely related to vector fields. Mathematically speaking streamlines are\u00a0continuous lines whose tangent at each point is given by a vector field. 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