Interpolation Exercise 1



In [1]:

    
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
import numpy as np



In [2]:

    
from scipy.interpolate import interp1d

2D trajectory interpolation

The file trajectory.npz contains 3 Numpy arrays that describe a 2d trajectory of a particle as a function of time:

t which has discrete values of time t[i].
x which has values of the x position at those times: x[i] = x(t[i]).
x which has values of the y position at those times: y[i] = y(t[i]).

Load those arrays into this notebook and save them as variables x, y and t:



In [13]:

    
trajectory = np.load('trajectory.npz')
x = trajectory['x']
y = trajectory['y']
t = trajectory['t']



In [14]:

    
assert isinstance(x, np.ndarray) and len(x)==40
assert isinstance(y, np.ndarray) and len(y)==40
assert isinstance(t, np.ndarray) and len(t)==40

Use these arrays to create interpolated functions $x(t)$ and $y(t)$. Then use those functions to create the following arrays:

newt which has 200 points between $\{t_{min},t_{max}\}$.
newx which has the interpolated values of $x(t)$ at those times.
newy which has the interpolated values of $y(t)$ at those times.



In [16]:

    
newt = np.linspace(min(t),max(t),200)
sin_approx = interp1d(t, x, kind='cubic')
sin_approx2 = interp1d(t, x, kind='cubic')
newx = sin_approx(newt)
newy = sin_approx(newt)



In [17]:

    
assert newt[0]==t.min()
assert newt[-1]==t.max()
assert len(newt)==200
assert len(newx)==200
assert len(newy)==200

Make a parametric plot of $\{x(t),y(t)\}$ that shows the interpolated values and the original points:

For the interpolated points, use a solid line.
For the original points, use circles of a different color and no line.
Customize you plot to make it effective and beautiful.



In [32]:

    
plt.plot(newx, newy, marker='o', linestyle='', label='original data')
plt.plot(newx, newy, marker='.', label='interpolated');
plt.legend(loc=4);
plt.xlabel('$x(t)$')
plt.ylabel('$y(t)$');
plt.xlim(-0.7, 0.9)
plt.ylim(-0.7,0.9)









    Out[32]:





(-0.7, 0.9)



In [ ]:

    
assert True # leave this to grade the trajectory plot