PlotBasics

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The Three Ways to plot

There are three ways to go about plotting in matplotlib.

1. You can use the pylab environment

2. You can use the matplotlib.pyplot environment, with plotting commands and functions.

3. You can define plot objects, and then use the pyplot methods on those objects.

The last way gives you most control, but the other two are somewhat easier. The examples below use the second way.

Supernova Cosmology Example

1. We will import the pyplot and numpy packages

import numpy as np
import matplotlib.pyplot as plt

Reading Data from the Web

We are going to download data from the internet so we will import the urllib package

import urllib

Begin by reading the Union2 SN cosmology data from LBL, because they are fun.

SN_list = []          
z_array = np.array([])
mod_array = np.array([])
moderr_array = np.array([])
f = urllib.urlopen('http://supernova.lbl.gov/Union/figures/SCPUnion2_mu_vs_z.txt')
for line in f:
    if line[0] == '#': continue    # Ignore anything that starts with a #
    SN, z, mod, moderr = line.split()
    SN_list.append(SN)
    z_array = np.append(z_array,np.float64(z))
    mod_array = np.append(mod_array,np.float64(mod))
    moderr_array = np.append(moderr_array,np.float64(moderr))   
f.close()

Simple Plots

Now to the plotting. First we close whatever windows we might have:

plt.close()

Now let us plot some points:

plt.plot(z_array, mod_array)

Notice it is a mess; by default it connects the lines. We will close it and start over.

plt.close()

We clearly need a title and axes labels

plt.title("Union2 SN Cosmology Data")
plt.xlabel('z', fontsize=20)

But we want a Greek Letter, so we can put some LaTeX syle code with the r command:

plt.ylabel(r'$\mu$', fontsize=20)

Now we add a format string to the plot command:

plt.plot(z_array, mod_array,'ro')

Now this looks better, we have red circles. The ro string, is what told it to give us red circles. A b. would give us blue dots, or ys would give yellow squares.

Plotting with Error Bars

Now let's plot with the error bars

plt.close() 
plt.errorbar(z_array, mod_array, yerr=moderr_array, fmt='.')

And putting labels and colors we can do:

plt.xlabel(r'$z$', fontsize=20)
plt.ylabel(r'$\mu=m-M$', fontsize=20)
plt.title("Union2 SN Cosmology Data")
plt.errorbar(z_array, mod_array, yerr=moderr_array, fmt='.', capsize=0,
    elinewidth=1.0, ecolor=(0.6,0.0,1.0), color='green' )

Notice that colors can be specified with a fmt or a color keyword. With the color keyword they can be given via an RGB tuple, an RGB hex code, a name, or a single number between '0.0' and '1.0' for gray scale.

Plotting Histograms

Now there are a lot of points, so let's figure our what our distribution is in z:

plt.hist(z_array, 25)
# And slap a label on it
plt.xlabel(r'$z$', fontsize=20)

plt.close()

Plotting With Asymmetrical Error Bars

Now, let's find the "real" distance from the distance modulus. To do this we will define a function and a constant.

def distance_Mpc(m,z):
    return 0.00001 * (10**(m/5)) / (1.0 + z)
c = 299792.458    # km/s

So the distance can be found with:

d_array = distance_Mpc(mod_array,z_array)

Now we will calculate the error bars in the distance, both ways:

d_error_plus  = ( distance_Mpc((mod_array+moderr_array),z_array) - d_array )
d_error_minus = ( d_array - distance_Mpc((mod_array-moderr_array),z_array) )

And plot the graph with asymetrical horizontal error bars, and labels. Notice the different color formats that can be used.

plt.close()
plt.errorbar(z_array, d_array, yerr=(d_error_minus,d_error_plus), fmt='s',
    capsize=5, elinewidth=1.0, color=(0.4,0.0,1.0), ecolor='aqua', barsabove=True)
plt.xlabel('z', fontsize=15, color='0.0')
plt.ylabel('Distance (Mpc)', fontsize=15, color='g')
plt.title("Union2 SN Cosmology Data", color=(0.4,0.0,1.0))

Plotting With a Second Axis

Now we will make a plot with both Mpc and Gly on the same plot.

First we read the existing plot limits into an array

axes1_range = np.array( plt.axis() )

We want to use the same x-axis so we use the function.

plt.twinx()   #  This swaps the Y axis
plt.ylabel('Distance in Billions of Light Years', fontsize=15, color='purple')

Now we will make the second axes the right scale.

axes2_range = axes1_range.copy()  # Don't forget, we need to make a copy of it.
axes2_range[2:4] = 3.26*axes1_range[2:4]/1000.0  # set second y-axis to Gly  
plt.axis(axes2_range)

plt.close()

Plotting With a Multiple Axes and Annotation

Now imagine we want to make a plot for the public, so we need to find the distance in light years. Also, we want to plot cz.

plt.errorbar(3.26*d_array/1000.0, c*z_array,
    xerr=(3.26*d_error_minus/1000.0, 3.26*d_error_plus/1000.0),
    fmt='.', capsize=0, elinewidth=1.0, color=(0.4,0.0,1.0), ecolor='aqua',
    barsabove=False, zorder=2)

Now we make room for each axis

plt.subplots_adjust(right=0.875, top=0.8)

And plot the labels.

plt.ylabel('cz (km/s)', fontsize=15, color='aqua')
plt.xlabel('Distance (Billions of Light Years)', fontsize=15, color='aqua')

Now, you want to make the plot also include z and Mpc, so we need to convert our axes:

axes1_range = np.array( plt.axis() ) # get the existing axes and convert to an array
print(axes1_range)
axes2_range = axes1_range.copy()  # Don't forget, we need to make a copy of it.
axes2_range[0:2] = 1000*axes1_range[0:2]/3.26  # set second x-axis to Mpc  
axes2_range[2:4] = axes1_range[2:4]/c # set second y-axis to z
print(axes2_range)

Now we switch to the second axis

plt.twinx()   #  This swaps the Y axis
plt.ylabel('z', fontsize=15, color='r')  #  I am not sure why this has to be before plt.twiny()
plt.twiny()   #  This swaps the X axis 
plt.xlabel('Distance (Mpc)', fontsize=15, color='purple')
plt.title("Union2 SN Cosmology Data", color=(0.4,0.0,1.0), x=0.5, y=1.15)
plt.axis(axes2_range)

And we will put Hubble's law on it just for fun. Using the zorder keyword, we can make sure the line is plotted on top on the points.

H = 70.8  # km/s/Mpc
z_limits = np.array([0.0,1.0])  #  Plot is relative to the new axes
d_limits = z_limits*c/H
plt.plot(d_limits,z_limits, 'y-', zorder=1)

Notice that the second plot rescaled the axes, which is a problem for this plot, because both x axes and both y axes represent the same physical quantity. So we reissue the axis function:

plt.axis(axes2_range)  # this needs to be after the plot command

Now we will put an arrow on the graph and a label.

plt.annotate("Hubble's law", xy=(d_limits[1],z_limits[1]), color='y', zorder=3, 
    xytext=(1.2*d_limits[1],1.2*z_limits[1]), arrowprops=dict(facecolor='y', shrink=0.05))

Saving as a PDF

We can save a file as a PDF or most any other common format this way:

plt.savefig('Union2_plot.pdf', format="pdf", transparent=True, bbox_inches='tight')

It is important to note that some file formats are raster graphics and others are scalable vector graphics.

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Magnetic Needle Example

Here we show another plotting example, where we calculate the magnetic field surrounding a thin magnetic needle.

First we import matplolib and numpy:

import numpy as np
import matplotlib.pyplot as plt

Next we define the grid size, length or needle, and the size of circular masks around the poles.

N = 20      # Size of NxN array.
R = 0.25    # Size of the two circular masks.
l = 1.0     # Length of the rod.
l2 = l/2.0  # Saves on typing l/2.0 all the time.

Now we will define min and max in the cylindrically radial coordinate (s), and the axial coordinate (z).

# Define s and z
s_delt = 1.0/(N/2)
s_max =  1*(1.0 + 0.5*s_delt)
s_min =  -1*(1.0 - 0.5*s_delt)
# Square Axes
z_delt = 1.0/(N/2)
z_min = -1*(1.0 - 0.5*z_delt)
z_max =  1*(1.0 + 0.5*z_delt)

Now we will find the 1-D space. Notice that it would have been slightly easier to use linspace here instead.

s_space_1D = np.arange(s_min,s_max,s_delt)
z_space_1D = np.arange(z_min,z_max,z_delt)

Now we define a 2-D grid of s and z values:

s, z = np.meshgrid(s_space_1D,z_space_1D)

And of course we find the magnetic field:

# Find each term in turn
Bs = 1.0/np.sqrt(s**2 + (z - l2)**2) - 1.0/np.sqrt(s**2 + (z + l2)**2)
Bs = Bs + (z + l2)**2 / (s**2 + (z + l2)**2)**1.5 - (z - l2)**2 / (s**2 + (z - l2)**2)**1.5
Bs = (1.0/(l*s))*Bs
Bz = (z + l2) / (s**2 + (z + l2)**2)**1.5 - (z - l2) / (s**2 + (z - l2)**2)**1.5
Bz = (-1.0/l)*Bz

Now we make and apply the mask to cover both poles:

mask = np.logical_or(np.sqrt(s**2 + (z-l2)**2) < R,np.sqrt(s**2 + (z+l2)**2) < R)
Bs = np.ma.masked_array(Bs, mask)
Bz = np.ma.masked_array(Bz, mask)

Now we can make an arrow plot of the magnetic field:

plt.close()
plt.box(on='on')
plt.axis('scaled')
plt.axis((-1.1, 1.1, -1.1, 1.1))
plt.title('Magnetic Field Surrounding a Thin Magnetic Needle')
plt.xlabel(r'$\bf s$', fontsize=20)
plt.ylabel(r'$\bf z$', fontsize=20)
plt.quiver(s,z,Bs,Bz,pivot='middle')
plt.savefig("Magnetic_field_of_thin_magnet.pdf", format="pdf", transparent=True, 
    bbox_inches='tight')

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Vector Potential Example

Here we show another plotting example, where we calculate the vector potential surrounding a square 2D magnet. We will also find the magnetic field, and plot a contour plot with a vector plot overlaid.

In this example, the vector potential has only a z component, so we will plot it as a filled contour plot, with the location of the magnet and the magnetic field vectors overlaid.

Calculating the Vector Potential and Magnetic Field

First we need to import the packages:

import numpy as np
import matplotlib.pyplot as plt

Now, we define the magnet dimensions and sizes of arrays

L = 1.0  # Length of Magnet
W = 1.0  # Width of Magnet
Nplt = 16   # Size of output magnetic field quiver plot
skip = 5  # Plot every skip point in each direction
smooth = 4   #Median of points for finding derivatives for added robustness
N = skip*(Nplt + 2) # Make NxN array for the total calculation
n = 30  # Size of nxn array that bounds the magnet

Next we will define the X and Y ranges

# Define Maximum and minimum area of the magnet
X_max = W/2.0
X_min = -1*W/2.0
Y_max = L/2.0
Y_min = -1*L/2.0
# Define the X and Y 1 D arrays
X = np.linspace(X_min,X_max,num=n)
Y = np.linspace(Y_min,Y_max,num=n)
# And corresponing 2D arrays
XX, YY = np.meshgrid(X,Y)

# Define Maximum and minimum area of the calculation
scale =( N + 2*smooth )/(N.__float__() - 1.0)
x_max = 1.0*scale
x_min = -1.0*scale
y_max = 1.0*scale
y_min = -1.0*scale
# Define the x and y 2D arrays for whole area
x, y = np.meshgrid(np.linspace(x_min,x_max,num=N),np.linspace(y_min,y_max,num=N))

# Define dx, dy, dX, dY, dR
dx = x[N/2+1,N/2+1] - x[N/2,N/2]
dy = y[N/2+1,N/2+1] - y[N/2,N/2]
dX = X[n/2+1] - X[n/2]
dY = Y[n/2+1] - Y[n/2]
dR = np.sqrt(dX**2 + dY**2)

Now we define the magnetization array as a constant vector in the y-direction.

My = 1.0 * (np.abs(YY) >= L/2.0)*(np.abs(YY) >= W/2.0)
Mx = 0.0 * My  
M = np.sqrt(Mx**2 + My**2)

Now, we find the differential vector potential (dAz) as a function of the location of the magnetic dipole element (X,Y) and the position of interest (x,y).

First we initialize some variables:

dAz = np.zeros((n,n,N,N))   # initialize dA as an n x n x N x N array
rR2  =  np.zeros((N,N))  # initialize N x N matrixe
Az   =  np.zeros((N,N))
pi = 3.14159265359

We need to make sure that we are ignoring distances smaller than the gridsize, so we define a minimum distance:

dR2  = dR**2   #  The X Y diagonal gridsize spacing squared
rR2max = (x_max-x_min+X_max-X_min)**2 + (y_max-y_min+Y_max-Y_min)**2

Now remember a 4-D array is simply a 2D array of 2D arrays, so we iterate over each position on the magnet.

i = 0
while i < n:  # The X Loop
    j = 0
    while j < n:  # The Y Loop
        rR2 = (x-X[i])**2 + (y-Y[j])**2
        rR2 = rR2.clip(2*dR2, 10*rR2max)  #  Clip at two grid spacing, and large distances
        dAz[j,i,:,:] = (0.5/pi) * ( Mx[j,i]*(y-Y[j]) - My[j,i]*(x - X[i]) ) / rR2
        j += 1
    i += 1
# End Looping

Now we numerically integrate over the magnet, to find the vector potential, using the trapezoid method. Remember dAz is a 4D array, but Az is a 2D array.

Az = np.trapz(np.trapz(dAz, Y, axis=0), X, axis=0)

Now that we have the vector potential, we need to take the curl to find the magnetic field. We will take multiple derivatives, for added robustness.

Bx   =  np.zeros((N,N))  # Initiate Bx and By
By   =  np.zeros((N,N))
Dx   =  np.zeros(smooth-2)   # Initiate the Dx and Dy arrays 
Dy   =  np.zeros(smooth-2)
i = smooth
while i < N-smooth:  # The x Loop
    j = smooth
    while j < N-smooth:  # The y Loop
        k = 2
        while k < smooth:   #  This loop is to add robustness by calculating multiple derivatives
            Dx[k-2] =    (Az[j,i+k+1] - Az[j,i-k])/(x[j,i+k+1] - x[j,i-k])
            Dy[k-2] =    (Az[j+k+1,i] - Az[j-k,i])/(y[j+k+1,i] - y[j-k,i])
            k += 1
        Bx[j,i] = np.median(Dy)
        By[j,i] = -1*np.median(Dx)
        j += 1
    i += 1
# End Looping

Now we will make smaller arrays to plot the magnetic field.

step = skip
start = (N-skip*Nplt)/2 + 2
stop = (N+skip*(Nplt-1))/2 + 1
BpltX   =  Bx[start:stop:step,start:stop:step]
BpltY   =  By[start:stop:step,start:stop:step]
xplt    =  x[start:stop:step,start:stop:step]
yplt    =  y[start:stop:step,start:stop:step]

Plotting

First we close and open a plot window, and set the axes and lables.

plt.close()
plt.box(on='on')
plt.axis('scaled')
plt.axis((-1.1, 1.1, -1.1, 1.1))    
plt.xlabel(r'$\bf x$', fontsize=20)
plt.ylabel(r'$\bf y$', fontsize=20)
TITLE ="A 2D Square Magnet"
plt.title(TITLE, weight='bold')

First we will make a grayscale plot of the vector potential: Had we instead had wanted to make a continuum grayscale plot, we could do:

plt.gray()
plt.pcolor(x,y,Az, alpha=1.0, zorder=0)

Now we will plot a box to represent the magnet. Alpha represents the opacity, and zorder represents what layer this represents.

box1x = (-1*L/2.0,L/2.0,L/2.0,-1*L/2.0)
box1y = (-1*W/2.0,-1*W/2.0,W/2.0,W/2.0)
plt.fill(box1x,box1y, facecolor='silver', edgecolor='None', alpha=0.5 , zorder=2)

Now, we will make a colorful filled contour plot of the vector potential, and plot it underneath the box, but on top of boring gray scale plot we made. We will also include a colorbar.

plt.spectral()
plt.contourf(x,y,Az,20, alpha=1.0, zorder=1)
plt.colorbar(orientation='vertical')
plt.figtext(0.92, 0.35, 'The Vector Potential', rotation=-90, weight='bold')

Now, let's plot white magnetic field vectors on top of the vector potential plot. We will also make sure that the longest arrow length is the grid spacing.

scale = np.max(np.sqrt(BpltX**2 + BpltY**2)) * Nplt  # The bigger the scale, the shorter the arrow
plt.quiver(xplt,yplt,BpltX,BpltY,pivot='middle',units='height', scale=scale, 
    zorder=4, color='white')

And if you want to save this plot, you can do so:

filename = TITLE.replace(' ','_') + "plot_3" + ".pdf"
plt.savefig(filename, format="pdf", transparent=True, \
    bbox_inches='tight')
print("Saved file as: " + filename)

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