Lektion 10



In [1]:

    
from sympy import *
init_printing()
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from IPython.display import display

Ebene parametrische Plots



In [2]:

    
tn = np.linspace(0, 2*np.pi, 550)
plt.plot(np.cos(7*tn), np.sin(11*tn));



In [3]:

    
plt.plot(np.cos(7*tn), np.sin(11*tn), 'g')
plt.plot(np.cos(7*tn+.1), np.sin(11*tn), 'r')
plt.axis('image');

Polarplot



In [4]:

    
tn = np.linspace(0, 2*np.pi, 250)
r = 2 + np.cos(tn)
phi = 2*tn
x = r*np.cos(phi)
y = r*np.sin(phi)
plt.plot(x, y)
plt.axis('equal')









    Out[4]:





$$\left ( -3.0, \quad 3.0, \quad -3.0, \quad 3.0\right )$$

Polarkoordinaten



In [5]:

    
x = Symbol('x')
y = Symbol('y')



In [6]:

    
x1 = 3*cos(pi/6)
y1 = 3*sin(pi/6)



In [7]:

    
atan(y1/x1)









    Out[7]:





$$\frac{\pi}{6}$$



In [8]:

    
x2 = 3*cos(-5*pi/6)
y2 = 3*sin(-5*pi/6)



In [9]:

    
atan(y2/x2)









    Out[9]:





$$\frac{\pi}{6}$$



In [10]:

    
atan2(y1, x1)









    Out[10]:





$$\frac{\pi}{6}$$



In [11]:

    
atan2(y2, x2)









    Out[11]:





$$- \frac{5 \pi}{6}$$



In [12]:

    
tn = np.linspace(-4, 4)
xn = np.cos(tn)
yn = np.sin(tn)
an1 = lambdify(x, atan(x), 'numpy')



In [13]:

    
an2 = lambdify((y,x), atan2(y,x), 'numpy')



In [14]:

    
plt.plot(tn, an1(yn/xn), 'r', label='atan')
plt.plot(tn, an2(yn, xn), 'g', label='atan2')
plt.legend(loc='upper center');

Polygonzüge



In [15]:

    
dreieck = [[0,0], [1,0], [0,1], [0,0]]
plt.plot([0,1,0,0], [0,0,1,0])
plt.axis(xmin=-.01,ymin=-.01);



In [16]:

    
def n_eck(n):
    x = [np.sin(2*np.pi*j/n) for j in range(n+1)]
    y = [np.cos(2*np.pi*j/n) for j in range(n+1)]
    return x, y



In [17]:

    
n_eck(3)









    Out[17]:





$$\left ( \left [ 0.0, \quad 0.866025403784, \quad -0.866025403784, \quad -2.44929359829e-16\right ], \quad \left [ 1.0, \quad -0.5, \quad -0.5, \quad 1.0\right ]\right )$$



In [18]:

    
for n in range(3, 9):
    plt.plot(*n_eck(n))  # <- Auspacken
plt.axis('image');

Ohne Auspacken:



In [24]:

    
for n in range(3, 9):
    x, y = n_eck(n)
    farbe = plt.cm.hsv(n/9)
    #for i in range(n-2):
    plt.plot(x, y, color=farbe, linewidth=2)
    plt.plot(x, y, 'o', color=farbe)
plt.axis('equal')









    Out[24]:





$$\left ( -1.0, \quad 1.0, \quad -1.0, \quad 1.0\right )$$



In [25]:

    
farbe









    Out[25]:





$$\left ( 1.0, \quad 0.0, \quad 0.742280060295, \quad 1.0\right )$$

Datenim- und export



In [26]:

    
H = Matrix(5, 5, [Rational(1, i+j+1) for i in range(5) for j in range(5)])
H









    Out[26]:





$$\left[\begin{matrix}1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5}\\\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6}\\\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7}\\\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8}\\\frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9}\end{matrix}\right]$$



In [27]:

    
str(H)









    Out[27]:





'Matrix([[1, 1/2, 1/3, 1/4, 1/5], [1/2, 1/3, 1/4, 1/5, 1/6], [1/3, 1/4, 1/5, 1/6, 1/7], [1/4, 1/5, 1/6, 1/7, 1/8], [1/5, 1/6, 1/7, 1/8, 1/9]])'



In [28]:

    
srepr(H)









    Out[28]:





'MutableDenseMatrix([[Integer(1), Rational(1, 2), Rational(1, 3), Rational(1, 4), Rational(1, 5)], [Rational(1, 2), Rational(1, 3), Rational(1, 4), Rational(1, 5), Rational(1, 6)], [Rational(1, 3), Rational(1, 4), Rational(1, 5), Rational(1, 6), Rational(1, 7)], [Rational(1, 4), Rational(1, 5), Rational(1, 6), Rational(1, 7), Rational(1, 8)], [Rational(1, 5), Rational(1, 6), Rational(1, 7), Rational(1, 8), Rational(1, 9)]])'



In [29]:

    
with open('bsp.txt', 'w') as fp:  # w: write
    fp.write(srepr(H) + '\n')



In [30]:

    
%less bsp.txt



In [31]:

    
H = []
H









    Out[31]:





$$\left [ \right ]$$



In [32]:

    
with(open('bsp.txt')) as fp:  # default ist lesen
    text = fp.readline()
text









    Out[32]:





'MutableDenseMatrix([[Integer(1), Rational(1, 2), Rational(1, 3), Rational(1, 4), Rational(1, 5)], [Rational(1, 2), Rational(1, 3), Rational(1, 4), Rational(1, 5), Rational(1, 6)], [Rational(1, 3), Rational(1, 4), Rational(1, 5), Rational(1, 6), Rational(1, 7)], [Rational(1, 4), Rational(1, 5), Rational(1, 6), Rational(1, 7), Rational(1, 8)], [Rational(1, 5), Rational(1, 6), Rational(1, 7), Rational(1, 8), Rational(1, 9)]])\n'



In [33]:

    
H = S(text)
H









    Out[33]:





$$\left[\begin{matrix}1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5}\\\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6}\\\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7}\\\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8}\\\frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9}\end{matrix}\right]$$



In [34]:

    
x = [Symbol('x_' + str(j+1)) for j in range(5)]



In [35]:

    
V = Matrix(5, 5, [x[j]**k for k in range(5) for j in range(5)])
V   # Vandermonde Matrix









    Out[35]:





$$\left[\begin{matrix}1 & 1 & 1 & 1 & 1\\x_{1} & x_{2} & x_{3} & x_{4} & x_{5}\\x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & x_{4}^{2} & x_{5}^{2}\\x_{1}^{3} & x_{2}^{3} & x_{3}^{3} & x_{4}^{3} & x_{5}^{3}\\x_{1}^{4} & x_{2}^{4} & x_{3}^{4} & x_{4}^{4} & x_{5}^{4}\end{matrix}\right]$$



In [36]:

    
with open('bsp.txt', 'a') as fp:   # a: append
    fp.write(srepr(V))



In [37]:

    
H = []
V = []
H, V









    Out[37]:





$$\left ( \left [ \right ], \quad \left [ \right ]\right )$$



In [38]:

    
with open('bsp.txt') as fp:
    H = S(fp.readline())
    V = S(fp.readline())



In [39]:

    
H, V









    Out[39]:





$$\left ( \left[\begin{matrix}1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5}\\\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6}\\\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7}\\\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8}\\\frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9}\end{matrix}\right], \quad \left[\begin{matrix}1 & 1 & 1 & 1 & 1\\x_{1} & x_{2} & x_{3} & x_{4} & x_{5}\\x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & x_{4}^{2} & x_{5}^{2}\\x_{1}^{3} & x_{2}^{3} & x_{3}^{3} & x_{4}^{3} & x_{5}^{3}\\x_{1}^{4} & x_{2}^{4} & x_{3}^{4} & x_{4}^{4} & x_{5}^{4}\end{matrix}\right]\right )$$



In [40]:

    
V.det().factor()









    Out[40]:





$$\left(x_{1} - x_{2}\right) \left(x_{1} - x_{3}\right) \left(x_{1} - x_{4}\right) \left(x_{1} - x_{5}\right) \left(x_{2} - x_{3}\right) \left(x_{2} - x_{4}\right) \left(x_{2} - x_{5}\right) \left(x_{3} - x_{4}\right) \left(x_{3} - x_{5}\right) \left(x_{4} - x_{5}\right)$$

Für den Datenaustausch empfiehlt sich pandas. Installation via

conda install pandas

Code in anderen Sprachen



In [41]:

    
x = Symbol('x')
y = Symbol('y')
f = log(x)*sqrt(x**2+3*y**2)
ddf = f.diff(x,y).simplify()
ddf









    Out[41]:





$$\frac{3 y}{x \left(x^{2} + 3 y^{2}\right)^{\frac{3}{2}}} \left(- x^{2} \log{\left (x \right )} + x^{2} + 3 y^{2}\right)$$



In [42]:

    
octave_code(ddf)









    Out[42]:





'3*y.*(-x.^2.*log(x) + x.^2 + 3*y.^2)./(x.*(x.^2 + 3*y.^2).^(3/2))'



In [43]:

    
ccode(ddf)









    Out[43]:





'3*y*(-pow(x, 2)*log(x) + pow(x, 2) + 3*pow(y, 2))/(x*pow(pow(x, 2) + 3*pow(y, 2), 3.0L/2.0L))'



In [46]:

    
print(fcode(ddf))









    



      3*y*(-x**2*log(x) + x**2 + 3*y**2)/(x*(x**2 + 3*y**2)**(3.0d0/
     @ 2.0d0))



In [48]:

    
print(latex(ddf))









    



\frac{3 y}{x \left(x^{2} + 3 y^{2}\right)^{\frac{3}{2}}} \left(- x^{2} \log{\left (x \right )} + x^{2} + 3 y^{2}\right)

Raumkurven



In [49]:

    
from mpl_toolkits.mplot3d import Axes3D



In [50]:

    
t = Symbol('t')
x = (2-cos(t/6))*cos(t)
y = (2-cos(t/6))*sin(t)
z = t/8
x, y, z









    Out[50]:





$$\left ( \left(- \cos{\left (\frac{t}{6} \right )} + 2\right) \cos{\left (t \right )}, \quad \left(- \cos{\left (\frac{t}{6} \right )} + 2\right) \sin{\left (t \right )}, \quad \frac{t}{8}\right )$$



In [51]:

    
xf = lambdify(t, x, 'numpy')
yf = lambdify(t, y, 'numpy')
zf = lambdify(t, z)



In [52]:

    
tn = np.linspace(-6*np.pi, 6*np.pi, 230)
xn = xf(tn)
yn = yf(tn)
zn = zf(tn)



In [53]:

    
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot(xn, yn, zn);



In [54]:

    
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
for i in range(len(tn)-1):
    ax.plot(xn[i:i+2], yn[i:i+2], zn[i:i+2], 
            color=plt.cm.hsv(float(i)/len(tn)),
            linewidth=5);

Trick aus http://stackoverflow.com/questions/15617207/line-colour-of-3d-parametric-curve-in-pythons-matplotlib-pyplot

Flächen im Raum



In [55]:

    
t = Symbol('t', real=True)
s = Symbol('s', real=True)



In [56]:

    
profil = (1-t**2)/2



In [57]:

    
x = t
y = cos(2*pi*s)*profil
z = sin(2*pi*s)*profil
w = cos(16*s)**4 * cos(16*t)**4  # Funktion mit Werten
                                 # in [0,1] für die Färbung
x, y, z









    Out[57]:





$$\left ( t, \quad \left(- \frac{t^{2}}{2} + \frac{1}{2}\right) \cos{\left (2 \pi s \right )}, \quad \left(- \frac{t^{2}}{2} + \frac{1}{2}\right) \sin{\left (2 \pi s \right )}\right )$$



In [58]:

    
xf = lambdify((s, t), x)
yf = lambdify((s, t), y, 'numpy')
zf = lambdify((s, t), z, 'numpy')
wf = lambdify((s, t), w, 'numpy')



In [59]:

    
tn = np.linspace(-1, 1, 201)
sn = np.linspace(-1, 1, 201)
S, T = np.meshgrid(sn, tn)



In [60]:

    
X = xf(S, T)
Y = yf(S, T)
Z = zf(S, T)
W = wf(S, T)



In [61]:

    
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_wireframe(X, Y, Z);



In [62]:

    
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X, Y, Z, rstride=1, cstride=1,
               linewidth=0, facecolors=plt.cm.viridis(1-W))
ax.auto_scale_xyz([-1,1], [-1,1], [-1,1]);

Vektorfelder



In [63]:

    
xn = np.linspace(-1, 1, 12)
yn = xn
X, Y = np.meshgrid(xn, yn)
U = X/np.sqrt(X**2+Y**2)
V = Y/np.sqrt(X**2+Y**2)



In [64]:

    
plt.quiver(X, Y, U, V);



In [65]:

    
N = (X**2+Y**2)**(0.3)  # Längenunterschiede werden vermindert



In [66]:

    
plt.quiver(X, Y, Y/N, -X/N, color='red')
plt.axis('image');



In [ ]: