Anaconda Enterprise: Documentation

# quiver.py

    from __future__ import division

    import numpy as np

    from bokeh.layouts import gridplot
    from bokeh.plotting import figure, show, output_file

    def streamlines(x, y, u, v, density=1):
        ''' Return streamlines of a vector flow.

        * x and y are 1d arrays defining an *evenly spaced* grid.
        * u and v are 2d arrays (shape [y,x]) giving velocities.
        * density controls the closeness of the streamlines. For different
          densities in each direction, use a tuple or list [densityx, densityy].

        '''

        ## Set up some constants - size of the grid used.
        NGX = len(x)
        NGY = len(y)

        ## Constants used to convert between grid index coords and user coords.
        DX = x[1]-x[0]
        DY = y[1]-y[0]
        XOFF = x[0]
        YOFF = y[0]

        ## Now rescale velocity onto axes-coordinates
        u = u / (x[-1]-x[0])
        v = v / (y[-1]-y[0])
        speed = np.sqrt(u*u+v*v)
        ## s (path length) will now be in axes-coordinates, but we must
        ## rescale u for integrations.
        u *= NGX
        v *= NGY
        ## Now u and v in grid-coordinates.

        NBX = int(30*density)
        NBY = int(30*density)
        blank = np.zeros((NBY,NBX))

        bx_spacing = NGX/float(NBX-1)
        by_spacing = NGY/float(NBY-1)

        def blank_pos(xi, yi):
            return int((xi / bx_spacing) + 0.5), \
                   int((yi / by_spacing) + 0.5)

        def value_at(a, xi, yi):
            if type(xi) == np.ndarray:
                x = xi.astype(np.int)
                y = yi.astype(np.int)
            else:
                x = np.int(xi)
                y = np.int(yi)
            a00 = a[y,x]
            a01 = a[y,x+1]
            a10 = a[y+1,x]
            a11 = a[y+1,x+1]
            xt = xi - x
            yt = yi - y
            a0 = a00*(1-xt) + a01*xt
            a1 = a10*(1-xt) + a11*xt
            return a0*(1-yt) + a1*yt

        def rk4_integrate(x0, y0):
            ## This function does RK4 forward and back trajectories from
            ## the initial conditions, with the odd 'blank array'
            ## termination conditions. TODO tidy the integration loops.

            def f(xi, yi):
                dt_ds = 1./value_at(speed, xi, yi)
                ui = value_at(u, xi, yi)
                vi = value_at(v, xi, yi)
                return ui*dt_ds, vi*dt_ds

            def g(xi, yi):
                dt_ds = 1./value_at(speed, xi, yi)
                ui = value_at(u, xi, yi)
                vi = value_at(v, xi, yi)
                return -ui*dt_ds, -vi*dt_ds

            check = lambda xi, yi: xi>=0 and xi<NGX-1 and yi>=0 and yi<NGY-1

            bx_changes = []
            by_changes = []

            ## Integrator function
            def rk4(x0, y0, f):
                ds = 0.01 #min(1./NGX, 1./NGY, 0.01)
                stotal = 0
                xi = x0
                yi = y0
                xb, yb = blank_pos(xi, yi)
                xf_traj = []
                yf_traj = []
                while check(xi, yi):
                    # Time step. First save the point.
                    xf_traj.append(xi)
                    yf_traj.append(yi)
                    # Next, advance one using RK4
                    try:
                        k1x, k1y = f(xi, yi)
                        k2x, k2y = f(xi + .5*ds*k1x, yi + .5*ds*k1y)
                        k3x, k3y = f(xi + .5*ds*k2x, yi + .5*ds*k2y)
                        k4x, k4y = f(xi + ds*k3x, yi + ds*k3y)
                    except IndexError:
                        # Out of the domain on one of the intermediate steps
                        break
                    xi += ds*(k1x+2*k2x+2*k3x+k4x) / 6.
                    yi += ds*(k1y+2*k2y+2*k3y+k4y) / 6.
                    # Final position might be out of the domain
                    if not check(xi, yi): break
                    stotal += ds
                    # Next, if s gets to thres, check blank.
                    new_xb, new_yb = blank_pos(xi, yi)
                    if new_xb != xb or new_yb != yb:
                        # New square, so check and colour. Quit if required.
                        if blank[new_yb,new_xb] == 0:
                            blank[new_yb,new_xb] = 1
                            bx_changes.append(new_xb)
                            by_changes.append(new_yb)
                            xb = new_xb
                            yb = new_yb
                        else:
                            break
                    if stotal > 2:
                        break
                return stotal, xf_traj, yf_traj

            integrator = rk4

            sf, xf_traj, yf_traj = integrator(x0, y0, f)
            sb, xb_traj, yb_traj = integrator(x0, y0, g)
            stotal = sf + sb
            x_traj = xb_traj[::-1] + xf_traj[1:]
            y_traj = yb_traj[::-1] + yf_traj[1:]

            ## Tests to check length of traj. Remember, s in units of axes.
            if len(x_traj) < 1: return None
            if stotal > .2:
                initxb, inityb = blank_pos(x0, y0)
                blank[inityb, initxb] = 1
                return x_traj, y_traj
            else:
                for xb, yb in zip(bx_changes, by_changes):
                    blank[yb, xb] = 0
                return None

        ## A quick function for integrating trajectories if blank==0.
        trajectories = []
        def traj(xb, yb):
            if xb < 0 or xb >= NBX or yb < 0 or yb >= NBY:
                return
            if blank[yb, xb] == 0:
                t = rk4_integrate(xb*bx_spacing, yb*by_spacing)
                if t is not None:
                    trajectories.append(t)

        ## Now we build up the trajectory set. I've found it best to look
        ## for blank==0 along the edges first, and work inwards.
        for indent in range((max(NBX,NBY))//2):
            for xi in range(max(NBX,NBY)-2*indent):
                traj(xi+indent, indent)
                traj(xi+indent, NBY-1-indent)
                traj(indent, xi+indent)
                traj(NBX-1-indent, xi+indent)

        xs = [np.array(t[0])*DX+XOFF for t in trajectories]
        ys = [np.array(t[1])*DY+YOFF for t in trajectories]

        return xs, ys

    xx = np.linspace(-3, 3, 100)
    yy = np.linspace(-3, 3, 100)

    Y, X = np.meshgrid(xx, yy)
    U = -1 - X**2 + Y
    V = 1 + X - Y**2
    speed = np.sqrt(U*U + V*V)
    theta = np.arctan(V/U)

    x0 = X[::2, ::2].flatten()
    y0 = Y[::2, ::2].flatten()
    length = speed[::2, ::2].flatten()/40
    angle = theta[::2, ::2].flatten()
    x1 = x0 + length * np.cos(angle)
    y1 = y0 + length * np.sin(angle)

    xs, ys = streamlines(xx, yy, U.T, V.T, density=2)

    cm = np.array(["#C7E9B4", "#7FCDBB", "#41B6C4", "#1D91C0", "#225EA8", "#0C2C84"])
    ix = ((length-length.min())/(length.max()-length.min())*5).astype('int')
    colors = cm[ix]

    p1 = figure(x_range=(-3,3 ), y_range=(-3, 3))
    p1.segment(x0, y0, x1, y1, color=colors, line_width=2)

    p2 = figure(x_range=p1.x_range, y_range=p1.y_range)
    p2.multi_line(xs, ys, color="#ee6666", line_width=2, line_alpha=0.8)

    output_file("vector.html", title="vector.py example")

    show(gridplot([[p1,p2]], plot_width=400, plot_height=400))  # open a browser